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Theorem inex1 3882
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 3868 . . 3 xy(y x ↔ (y A y B))
3 dfcleq 2031 . . . . 5 (x = (AB) ↔ y(y xy (AB)))
4 elin 3120 . . . . . . 7 (y (AB) ↔ (y A y B))
54bibi2i 216 . . . . . 6 ((y xy (AB)) ↔ (y x ↔ (y A y B)))
65albii 1356 . . . . 5 (y(y xy (AB)) ↔ y(y x ↔ (y A y B)))
73, 6bitri 173 . . . 4 (x = (AB) ↔ y(y x ↔ (y A y B)))
87exbii 1493 . . 3 (x x = (AB) ↔ xy(y x ↔ (y A y B)))
92, 8mpbir 134 . 2 x x = (AB)
109issetri 2558 1 (AB) V
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cin 2910
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918
This theorem is referenced by:  inex2  3883  inex1g  3884  inuni  3900  bnd2  3917  peano5  4264  ssimaex  5177  ofmres  5705  tfrexlem  5889
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