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Theorem inex1 3916
Description: Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
inex1.1 𝐴 ∈ V
Assertion
Ref Expression
inex1 (𝐴𝐵) ∈ V

Proof of Theorem inex1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inex1.1 . . . 4 𝐴 ∈ V
21zfauscl 3902 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
3 dfcleq 2048 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
4 elin 3151 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
54bibi2i 220 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
65albii 1373 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
73, 6bitri 177 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
87exbii 1510 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
92, 8mpbir 138 . 2 𝑥 𝑥 = (𝐴𝐵)
109issetri 2579 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wal 1255   = wceq 1257  wex 1395  wcel 1407  Vcvv 2572  cin 2941
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 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-in 2949
This theorem is referenced by:  inex2  3917  inex1g  3918  inuni  3934  bnd2  3951  peano5  4346  ssimaex  5259  ofmres  5788  tfrexlem  5976
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