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Theorem inex1 4030
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 4016 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
3 dfcleq 2109 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
4 elin 3227 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
54bibi2i 226 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
65albii 1429 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
73, 6bitri 183 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
87exbii 1567 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
92, 8mpbir 145 . 2 𝑥 𝑥 = (𝐴𝐵)
109issetri 2667 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cin 3038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045
This theorem is referenced by:  inex2  4031  inex1g  4032  inuni  4048  bnd2  4065  peano5  4480  ssimaex  5448  ofmres  6000  tfrexlem  6197  elrest  12022  epttop  12154  tgrest  12233  resttopon  12235  restco  12238  cnrest2  12300  cnptopresti  12302  cnptoprest  12303  cnptoprest2  12304  txrest  12340
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