ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inex1 Structured version   GIF version

Theorem inex1 3863
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 3849 . . 3 xy(y x ↔ (y A y B))
3 dfcleq 2016 . . . . 5 (x = (AB) ↔ y(y xy (AB)))
4 elin 3101 . . . . . . 7 (y (AB) ↔ (y A y B))
54bibi2i 216 . . . . . 6 ((y xy (AB)) ↔ (y x ↔ (y A y B)))
65albii 1339 . . . . 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 1478 . . 3 (x x = (AB) ↔ xy(y x ↔ (y A y B)))
92, 8mpbir 134 . 2 x x = (AB)
109issetri 2540 1 (AB) V
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1226   = wceq 1228  wex 1362   wcel 1374  Vcvv 2533  cin 2891
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-in 2899
This theorem is referenced by:  inex2  3864  inex1g  3865  inuni  3881  bnd2  3898  peano5  4246  ssimaex  5157  ofmres  5684  tfrexlem  5868
  Copyright terms: Public domain W3C validator