Theorem bj-ideqg1ALT 37692

Description: Alternate proof of bj-ideqg1 using brabga 5516 instead of the "unbounded" version bj-brab2a1 37676 or brab2a 5752. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 37684 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 5811	. . . 4 ⊢ Rel I
2	1	brrelex12i 5714	. . 3 ⊢ (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	adantl 486	. 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		elex 3484	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
5	4	adantr 485	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
6		eleq1 2857	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
7	6	biimparc 484	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑊)
8	7	elexd 3486	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
9	5, 8	jaoian 971	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
10		eleq1 2857	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
11	10	biimpac 483	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉)
12	11	elexd 3486	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13		elex 3484	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
14	13	adantr 485	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
15	12, 14	jaoian 971	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
16	9, 15	jca 520	. 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17		eqeq12 2786	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
18		df-id 5554	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
19	17, 18	brabga 5516	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
20	3, 16, 19	pm5.21nd 813	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  Vcvv 3463   class class class wbr 5110   I cid 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666

This theorem is referenced by: (None)