Theorem bj-ideqb 37203

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 5765	. . 3 ⊢ Rel I
2	1	brrelex1i 5670	. 2 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
3		inex1g 5255	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
4		bj-ideqg 37201	. . 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 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   class class class wbr 5089   I cid 5508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621

This theorem is referenced by: (None)