Theorem nd2 10574

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)

Assertion

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

Proof of Theorem nd2

Step	Hyp	Ref	Expression
1		elirrv 9560	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2102	. . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦)
3	1	nfnth 1832	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ2 2158	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2534	. . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 221	. . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧)
7	1, 6	mto 200	. 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦
8		axc11 2462	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦))
9	7, 8	mtoi 202	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1568  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-13 2404  ax-sep 5258  ax-reg 9555

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is referenced by:  axrepnd  10580  axpownd  10587  axinfndlem1  10591  axacndlem4  10596