Theorem nd3 10602

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

Assertion

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

Proof of Theorem nd3

Step	Hyp	Ref	Expression
1		elirrv 9573	. . . 4 ⊢ ¬ 𝑥 ∈ 𝑥
2		elequ2 2160	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦))
3	1, 2	mtbii 329	. . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
4	3	sps 2223	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
5		sp 2221	. 2 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦)
6	4, 5	nsyl 141	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-sep 5255  ax-reg 9568

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  nd4  10603  axrepnd  10607  axpowndlem3  10612  axinfnd  10619  axacndlem3  10622  axacnd  10625