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Theorem ideqg 4694
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 reli 4672 . . . . 5 Rel I
21brrelex1i 4586 . . . 4 (𝐴 I 𝐵𝐴 ∈ V)
32adantl 275 . . 3 ((𝐵𝑉𝐴 I 𝐵) → 𝐴 ∈ V)
4 simpl 108 . . 3 ((𝐵𝑉𝐴 I 𝐵) → 𝐵𝑉)
53, 4jca 304 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
6 eleq1 2203 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
76biimparc 297 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
8 elex 2698 . . . 4 (𝐴𝑉𝐴 ∈ V)
97, 8syl 14 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
10 simpl 108 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
119, 10jca 304 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
12 eqeq1 2147 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
13 eqeq2 2150 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
14 df-id 4219 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1512, 13, 14brabg 4195 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
165, 11, 15pm5.21nd 902 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  Vcvv 2687   class class class wbr 3933   I cid 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-id 4219  df-xp 4549  df-rel 4550
This theorem is referenced by:  ideq  4695  ididg  4696  poleloe  4942
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