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Theorem inex1 3855
 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 3841 . . 3 xy(y x ↔ (y A y B))
3 dfcleq 2008 . . . . 5 (x = (AB) ↔ y(y xy (AB)))
4 elin 3095 . . . . . . 7 (y (AB) ↔ (y A y B))
54bibi2i 216 . . . . . 6 ((y xy (AB)) ↔ (y x ↔ (y A y B)))
65albii 1333 . . . . 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 1470 . . 3 (x x = (AB) ↔ xy(y x ↔ (y A y B)))
92, 8mpbir 134 . 2 x x = (AB)
109issetri 2534 1 (AB) V
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1222   = wceq 1224  ∃wex 1355   ∈ wcel 1367  Vcvv 2527   ∩ cin 2885 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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839 This theorem depends on definitions:  df-bi 110  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-v 2529  df-in 2893 This theorem is referenced by:  inex2  3856  inex1g  3857  inuni  3873  bnd2  3890  peano5  4237  ssimaex  5148  ofmres  5675  tfrexlem  5859
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