Theorem bj-ideqg1ALT 38291

Description: Alternate proof of bj-ideqg1 using brabga 5513 instead of the "unbounded" version bj-brab2a1 38275 or brab2a 5750. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 38283 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 5711	. . 3 ⊢ (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	adantl 487	. 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		elex 3475	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
5	4	adantr 486	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
6		eleq1 2852	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
7	6	biimparc 485	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑊)
8	7	elexd 3477	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
9	5, 8	jaoian 971	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
10		eleq1 2852	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
11	10	biimpac 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉)
12	11	elexd 3477	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13		elex 3475	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
14	13	adantr 486	. . . 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 521	. 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17		eqeq12 2780	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
18		df-id 5551	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
19	17, 18	brabga 5513	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
20	3, 16, 19	pm5.21nd 814	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106   I cid 5550

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-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663

This theorem is used by: (None)