Theorem bj-opelidb1ALT 37851

Description: Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-opelidb1ALT	⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-opelidb1ALT

Step	Hyp	Ref	Expression
1		df-br 5115	. . 3 ⊢ (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2		reli 5818	. . . 4 ⊢ Rel I
3	2	brrelex1i 5722	. . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
4	1, 3	sylbir 238	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I → 𝐴 ∈ V)
5		inex1g 5293	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
6		bj-opelid 37841	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7	5, 6	syl 18	. 2 ⊢ (𝐴 ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
8	4, 7	biadanii 834	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∩ cin 3907  ⟨cop 4600   class class class wbr 5114   I cid 5560

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

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-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673

This theorem is used by: (None)