Theorem distel 36742

Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5410 and elirrv 9615.) (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 5410	. . 3 ⊢ ∃𝑧 𝑥 ∈ 𝑧
2		df-ex 1813	. . . 4 ⊢ (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧)
3		nfnae 2466	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4		dveel1 2493	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
5	3, 4	nf5d 2321	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝑧)
6	5	nfnd 1891	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥 ∈ 𝑧)
7		elequ2 2162	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
8	7	notbid 321	. . . . . . 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 2439	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
11	10	notbid 321	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
12	2, 11	bitrid 286	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
13	1, 12	mpbii 236	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)
14		elirrv 9615	. . . . 5 ⊢ ¬ 𝑦 ∈ 𝑦
15		elequ1 2154	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
16	14, 15	mtbii 329	. . . 4 ⊢ (𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝑦)
17	16	alimi 1844	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥 ∈ 𝑦)
18	17	con3i 155	. 2 ⊢ (¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
19	13, 18	impbii 212	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-sep 5252  ax-pr 5395  ax-reg 9610

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by: (None)