Theorem bj-ideqb 37125

Description: Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)

Assertion

Ref	Expression
bj-ideqb	⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-ideqb

Step	Hyp	Ref	Expression
1		reli 5850	. . 3 ⊢ Rel I
2	1	brrelex1i 5756	. 2 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
3		inex1g 5337	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
4		bj-ideqg 37123	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
5	3, 4	syl 17	. 2 ⊢ (𝐴 ∈ V → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
6	2, 5	biadanii 821	1 ⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   class class class wbr 5166   I cid 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707

This theorem is referenced by: (None)