Q: Does every complete Boolean Algebra that is a proper class have a anti-chain that is a proper class? We work in Kelley-Morse class theory and assume global choice. Let B be a complete Boolean algebra that is a proper class (B is complete in the sense that every subclass of it has a minimal upper bound). Let's assume that B is NOT atomic. Do we know that B must have an anti-chain that is also a proper class?
I was trying to show that B has a maximal chain that is a proper class, which would produce an anti-chain that is a proper class. But I'm not clear if a maximal chain in B has to be a proper class.
