Theorem bj-ideqb 37651

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 5799	. . 3 ⊢ Rel I
2	1	brrelex1i 5703	. 2 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
3		inex1g 5275	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
4		bj-ideqg 37649	. . 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 831	1 ⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∩ cin 3903   class class class wbr 5100   I cid 5541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654

This theorem is referenced by: (None)