Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  clel3 GIF version

Theorem clel3 2679
 Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel3.1 𝐵 ∈ V
Assertion
Ref Expression
clel3 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel3
StepHypRef Expression
1 clel3.1 . 2 𝐵 ∈ V
2 clel3g 2678 . 2 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
31, 2ax-mp 7 1 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559 This theorem is referenced by:  unipr  3594
 Copyright terms: Public domain W3C validator