Theorem nd2 9610

Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)

Assertion

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

Proof of Theorem nd2

Step	Hyp	Ref	Expression
1		elirrv 8655	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2497	. . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦)
3	1	nfnth 1874	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ2 2157	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2553	. . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 208	. . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧)
7	1, 6	mto 188	. 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦
8		axc11 2464	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦))
9	7, 8	mtoi 190	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1627  [wsb 2047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033  ax-reg 8651

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ral 3064  df-rex 3065  df-v 3350  df-dif 3723  df-un 3725  df-nul 4061  df-sn 4314  df-pr 4316

This theorem is referenced by:  axrepnd  9616  axpownd  9623  axinfndlem1  9627  axacndlem4  9632