Theorem bj-ideqb 36496

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 5816	. . 3 ⊢ Rel I
2	1	brrelex1i 5722	. 2 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
3		inex1g 5309	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
4		bj-ideqg 36494	. . 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 819	1 ⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∩ cin 3939   class class class wbr 5138   I cid 5563

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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673

This theorem is referenced by: (None)