Theorem bj-ideqg1 37539

Description: For sets, the identity relation is the same thing as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ, 24-Dec-2023.)

TODO: delete once bj-ideqg 37532 is in the main section.

Assertion

Ref	Expression
bj-ideqg1	⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem bj-ideqg1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq12 2758	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
2		df-id 5516	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
3	1, 2	bj-brab2a1 37524	. 2 ⊢ (𝐴 I 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
4		simpr 486	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
5		elex 3454	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
6	5	a1d 25	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ V))
7		elex 3454	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
8		eleq1a 2836	. . . . . . 7 ⊢ (𝐵 ∈ V → (𝐴 = 𝐵 → 𝐴 ∈ V))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝐴 = 𝐵 → 𝐴 ∈ V))
10	6, 9	jaoi 864	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → 𝐴 ∈ V))
11		eleq1 2829	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
12	5, 11	syl5ibcom 247	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ V))
13	7	a1d 25	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝐴 = 𝐵 → 𝐵 ∈ V))
14	12, 13	jaoi 864	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → 𝐵 ∈ V))
15	10, 14	jcad 518	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
16	15	ancrd 557	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)))
17	4, 16	impbid2 228	. 2 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵))
18	3, 17	bitrid 285	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  Vcvv 3433   class class class wbr 5075   I cid 5515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627

This theorem is referenced by: (None)