Theorem distel 31407

Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4807 and elirrv 8448.) (Contributed by Scott Fenton, 15-Dec-2010.)

Assertion

Ref	Expression
distel	⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)

Proof of Theorem distel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		el 4807	. . 3 ⊢ ∃𝑧 𝑥 ∈ 𝑧
2		df-ex 1702	. . . 4 ⊢ (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧)
3		nfnae 2317	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4		dveel1 2369	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
5	3, 4	nf5d 2115	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝑧)
6	5	nfnd 1782	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥 ∈ 𝑧)
7		elequ2 2001	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
8	7	notbid 308	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑦))
9	8	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → (¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑦)))
10	3, 6, 9	cbvald 2276	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
11	10	notbid 308	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
12	2, 11	syl5bb 272	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
13	1, 12	mpbii 223	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)
14		elirrv 8448	. . . . 5 ⊢ ¬ 𝑦 ∈ 𝑦
15		elequ1 1994	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
16	14, 15	mtbii 316	. . . 4 ⊢ (𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝑦)
17	16	alimi 1736	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥 ∈ 𝑦)
18	17	con3i 150	. 2 ⊢ (¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
19	13, 18	impbii 199	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1478  ∃wex 1701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-reg 8441

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-pr 4151

This theorem is referenced by: (None)