Theorem mirimanofff

Description: Mirimanoff's paradox mirimanoff for founded sets. A set x is founded if every set containing x has a minimal element under membership. Compare with zfreg the defining condition of M in the hypothesis. (Contributed by Adrian Ducourtial, 28-Oct-2026)

	Ref	Expression
Hypothesis	mirimanofff.1	$$⊢M=\left\{x|\forall y\left(x\in y\to \exists z\in yz\cap y=\varnothing \right)\right\}$$
Assertion	mirimanofff	$$⊢M\notin \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		mirimanofff.1	$$⊢M=\left\{x|\forall y\left(x\in y\to \exists z\in yz\cap y=\varnothing \right)\right\}$$
2		rabab	$$⊢\left\{x\in \mathrm{V}|\forall y\left(x\in y\to \exists z\in yz\cap y=\varnothing \right)\right\}=\left\{x|\forall y\left(x\in y\to \exists z\in yz\cap y=\varnothing \right)\right\}$$
3	1 2	eqtr4i	$$⊢M=\left\{x\in \mathrm{V}|\forall y\left(x\in y\to \exists z\in yz\cap y=\varnothing \right)\right\}$$
4	3	mirimanofffr	$$⊢M\notin \mathrm{V}$$