Theorem nd3 10549

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 9556	. . . 4 ⊢ ¬ 𝑥 ∈ 𝑥
2		elequ2 2124	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦))
3	1, 2	mtbii 326	. . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
4	3	sps 2186	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
5		sp 2184	. 2 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦)
6	4, 5	nsyl 140	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-pr 5390  ax-reg 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-v 3452  df-un 3922  df-sn 4593  df-pr 4595

This theorem is referenced by:  nd4  10550  axrepnd  10554  axpowndlem3  10559  axinfnd  10566  axacndlem3  10569  axacnd  10572