Theorem ideqg 5876

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 5850	. . . 4 ⊢ Rel I
3	2	brrelex1i 5756	. . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
4	1, 3	anim12ci 613	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉))
5		eleq1 2832	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
6	5	biimparc 479	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉)
7	6	elexd 3512	. . 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 2744	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
11		eqeq2 2752	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
12		df-id 5593	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
13	10, 11, 12	brabg 5558	. 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 1537   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166   I cid 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707

This theorem is referenced by:  ideq  5877  ididg  5878  poleloe  6163  isof1oidb  7360  pltval  18402  tglngne  28576  tgelrnln  28656  opeldifid  32621  ideq2  38263  idinxpss  38268  inxpssidinxp  38272  idinxpssinxp  38273  cnvref5  38307  rnxrnidres  38357  cossid  38436  fourierdlem42  46070