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Theorem inex1 3887
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 3873 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
3 dfcleq 2034 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
4 elin 3123 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
54bibi2i 216 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
65albii 1359 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
73, 6bitri 173 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
87exbii 1496 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
92, 8mpbir 134 . 2 𝑥 𝑥 = (𝐴𝐵)
109issetri 2561 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wal 1241   = wceq 1243  wex 1381  wcel 1393  Vcvv 2554  cin 2913
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  ax-sep 3871
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 2556  df-in 2921
This theorem is referenced by:  inex2  3888  inex1g  3889  inuni  3905  bnd2  3922  peano5  4282  ssimaex  5195  ofmres  5724  tfrexlem  5909
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