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

Theorem ideqg 4557
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 4535 . . . . 5 Rel I
21brrelexi 4452 . . . 4 (𝐴 I 𝐵𝐴 ∈ V)
32adantl 271 . . 3 ((𝐵𝑉𝐴 I 𝐵) → 𝐴 ∈ V)
4 simpl 107 . . 3 ((𝐵𝑉𝐴 I 𝐵) → 𝐵𝑉)
53, 4jca 300 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
6 eleq1 2147 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
76biimparc 293 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
8 elex 2624 . . . 4 (𝐴𝑉𝐴 ∈ V)
97, 8syl 14 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
10 simpl 107 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
119, 10jca 300 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
12 eqeq1 2091 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
13 eqeq2 2094 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
14 df-id 4096 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1512, 13, 14brabg 4072 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
165, 11, 15pm5.21nd 861 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  Vcvv 2615   class class class wbr 3822   I cid 4091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420
This theorem is referenced by:  ideq  4558  ididg  4559  poleloe  4800
  Copyright terms: Public domain W3C validator