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Theorem ideqg 5717
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 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem ideqg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝐵𝑉𝐵𝑉)
2 reli 5693 . . . 4 Rel I
32brrelex1i 5603 . . 3 (𝐴 I 𝐵𝐴 ∈ V)
41, 3anim12ci 615 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
5 eleq1 2900 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
65biimparc 482 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
76elexd 3515 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
8 simpl 485 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
97, 8jca 514 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 eqeq1 2825 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqeq2 2833 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
12 df-id 5455 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1310, 11, 12brabg 5419 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
144, 9, 13pm5.21nd 800 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3495   class class class wbr 5059   I cid 5454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557
This theorem is referenced by:  ideq  5718  ididg  5719  poleloe  5986  isof1oidb  7071  pltval  17564  tglngne  26330  tgelrnln  26410  opeldifid  30343  ideq2  35559  idinxpss  35564  inxpssidinxp  35567  idinxpssinxp  35568  rnxrnidres  35643  cossid  35714  fourierdlem42  42427
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