Theorem nd4 10330

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
nd4	⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥)

Proof of Theorem nd4

Step	Hyp	Ref	Expression
1		nd3 10329	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑧 𝑦 ∈ 𝑥)
2	1	aecoms 2429	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-12 2174  ax-13 2373  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-reg 9312

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-sn 4567  df-pr 4569

This theorem is referenced by:  axrepnd  10334