ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ideqg GIF version

Theorem ideqg 4514
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 4492 . . . . 5 Rel I
21brrelexi 4411 . . . 4 (𝐴 I 𝐵𝐴 ∈ V)
32adantl 266 . . 3 ((𝐵𝑉𝐴 I 𝐵) → 𝐴 ∈ V)
4 simpl 106 . . 3 ((𝐵𝑉𝐴 I 𝐵) → 𝐵𝑉)
53, 4jca 294 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
6 eleq1 2116 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
76biimparc 287 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
8 elex 2583 . . . 4 (𝐴𝑉𝐴 ∈ V)
97, 8syl 14 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
10 simpl 106 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
119, 10jca 294 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
12 eqeq1 2062 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
13 eqeq2 2065 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
14 df-id 4057 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1512, 13, 14brabg 4033 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
165, 11, 15pm5.21nd 836 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574   class class class wbr 3791   I cid 4052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379
This theorem is referenced by:  ideq  4515  ididg  4516  poleloe  4751
  Copyright terms: Public domain W3C validator