Theorem nd2

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 1-Jan-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	nd2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑥 𝑧 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		elirrv	⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4	⊢ ( ∀ 𝑦 𝑧 ∈ 𝑦 → [ 𝑧 / 𝑦 ] 𝑧 ∈ 𝑦 )
3	1	nfnth	⊢ Ⅎ 𝑦 𝑧 ∈ 𝑧
4		elequ2	⊢ ( 𝑦 = 𝑧 → ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧 ) )
5	3 4	sbie	⊢ ( [ 𝑧 / 𝑦 ] 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧 )
6	2 5	sylib	⊢ ( ∀ 𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧 )
7	1 6	mto	⊢ ¬ ∀ 𝑦 𝑧 ∈ 𝑦
8		axc11	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑧 ∈ 𝑦 → ∀ 𝑦 𝑧 ∈ 𝑦 ) )
9	7 8	mtoi	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑥 𝑧 ∈ 𝑦 )