(CNN) — The same-sex couple who lost a major Supreme Court battle with a baker over his refusal on religious grounds to make them a wedding cake told CNN Tuesday they felt “shock and disappointment” over the decision.

“This has been going on for six years and to have it turn out this way was very disappointing,” Dave Mullins told CNN’s John Berman on “New Day.”

Mullins also took issue with baker Jack Phillips’ claim that he could not bake a wedding cake for their ceremony because it went against his religious beliefs. Phillips has also said he doesn’t create cakes for Halloween.

“Well, I think it’s important that we be clear on this, that Mr. Phillips would not sell us anything for our wedding,” Mullins said, “and as far as the example of the Halloween cake goes, you know, any human being could buy a Halloween cake, but only a same-sex couple needs a wedding for a so-called same-sex wedding.”

“I don’t feel like our case is about a cake,” Charlie Craig, Mullins’ husband, elaborated. “It’s about getting access to public accommodation.”

Still, Mullins nodded to some “silver linings” in the narrow ruling.

“I think the most important thing for us that we want people to understand is that the Colorado Anti-Discrimination Act is still fully in effect in Colorado,” he said. “Nothing has changed about that. It is still illegal to turn a gay couple away from a business just because of who they are.”

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