Theorem nd3

Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002)

	Ref	Expression
Assertion	nd3	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑧 𝑥 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		elirrv	⊢ ¬ 𝑥 ∈ 𝑥
2		elequ2	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦 ) )
3	1 2	mtbii	⊢ ( 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦 )
4	3	sps	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦 )
5		sp	⊢ ( ∀ 𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦 )
6	4 5	nsyl	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑧 𝑥 ∈ 𝑦 )