Theorem bj-ideqg1ALT 36697

Description: Alternate proof of bj-ideqg1 using brabga 5531 instead of the "unbounded" version bj-brab2a1 36681 or brab2a 5766. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 36689 is in the main section.

Assertion

Ref	Expression
bj-ideqg1ALT	⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem bj-ideqg1ALT

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

Step	Hyp	Ref	Expression
1		reli 5823	. . . 4 ⊢ Rel I
2	1	brrelex12i 5728	. . 3 ⊢ (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	adantl 480	. 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		elex 3482	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
5	4	adantr 479	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
6		eleq1 2813	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
7	6	biimparc 478	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑊)
8	7	elexd 3485	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
9	5, 8	jaoian 954	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
10		eleq1 2813	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
11	10	biimpac 477	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉)
12	11	elexd 3485	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13		elex 3482	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
14	13	adantr 479	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
15	12, 14	jaoian 954	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
16	9, 15	jca 510	. 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17		eqeq12 2742	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
18		df-id 5571	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
19	17, 18	brabga 5531	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
20	3, 16, 19	pm5.21nd 800	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  Vcvv 3463   class class class wbr 5144   I cid 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680

This theorem is referenced by: (None)