Theorem nd3 9290

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 8387	. . . 4 ⊢ ¬ 𝑥 ∈ 𝑥
2		elequ2 1991	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦))
3	1, 2	mtbii 315	. . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
4	3	sps 2043	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
5		sp 2041	. 2 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦)
6	4, 5	nsyl 134	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128

This theorem is referenced by:  nd4  9291  axrepnd  9295  axpowndlem3  9300  axinfnd  9307  axacndlem3  9310  axacnd  9313