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

Theorem ideqg 4410
 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 BA = B))

Proof of Theorem ideqg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 4388 . . . . 5 Rel I
21brrelexi 4307 . . . 4 (A I BA V)
32adantl 262 . . 3 ((B 𝑉 A I B) → A V)
4 simpl 102 . . 3 ((B 𝑉 A I B) → B 𝑉)
53, 4jca 290 . 2 ((B 𝑉 A I B) → (A V B 𝑉))
6 eleq1 2078 . . . . 5 (A = B → (A 𝑉B 𝑉))
76biimparc 283 . . . 4 ((B 𝑉 A = B) → A 𝑉)
8 elex 2539 . . . 4 (A 𝑉A V)
97, 8syl 14 . . 3 ((B 𝑉 A = B) → A V)
10 simpl 102 . . 3 ((B 𝑉 A = B) → B 𝑉)
119, 10jca 290 . 2 ((B 𝑉 A = B) → (A V B 𝑉))
12 eqeq1 2024 . . 3 (x = A → (x = yA = y))
13 eqeq2 2027 . . 3 (y = B → (A = yA = B))
14 df-id 4000 . . 3 I = {⟨x, y⟩ ∣ x = y}
1512, 13, 14brabg 3976 . 2 ((A V B 𝑉) → (A I BA = B))
165, 11, 15pm5.21nd 813 1 (B 𝑉 → (A I BA = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226   ∈ wcel 1370  Vcvv 2531   class class class wbr 3734   I cid 3995 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275 This theorem is referenced by:  ideq  4411  ididg  4412  poleloe  4647
 Copyright terms: Public domain W3C validator