Theorem ideqg 5836

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ideqg	⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem ideqg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝑉)
2		reli 5810	. . . 4 ⊢ Rel I
3	2	brrelex1i 5715	. . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
4	1, 3	anim12ci 614	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉))
5		eleq1 2823	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
6	5	biimparc 479	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉)
7	6	elexd 3488	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
8		simpl 482	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉)
9	7, 8	jca 511	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉))
10		eqeq1 2740	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
11		eqeq2 2748	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
12		df-id 5553	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
13	10, 11, 12	brabg 5519	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
14	4, 9, 13	pm5.21nd 801	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   class class class wbr 5124   I cid 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666

This theorem is referenced by:  ideq  5837  ididg  5838  poleloe  6125  isof1oidb  7322  pltval  18347  tglngne  28534  tgelrnln  28614  opeldifid  32585  ideq2  38330  idinxpss  38335  inxpssidinxp  38339  idinxpssinxp  38340  cnvref5  38374  rnxrnidres  38424  cossid  38503  fourierdlem42  46145