Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssex Unicode version

Theorem ssex 4065
 Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4046 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
Hypothesis
Ref Expression
ssex.1
Assertion
Ref Expression
ssex

Proof of Theorem ssex
StepHypRef Expression
1 df-ss 3084 . 2
2 ssex.1 . . . 4
32inex2 4063 . . 3
4 eleq1 2202 . . 3
53, 4mpbii 147 . 2
61, 5sylbi 120 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1331   wcel 1480  cvv 2686   cin 3070   wss 3071 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084 This theorem is referenced by:  ssexi  4066  ssexg  4067  inteximm  4074  funimaexglem  5206  tfrexlem  6231  elinp  7289  suplocexprlem2b  7529  negfi  11006  elcncf  12739  exmid1stab  13225  sbthom  13251
 Copyright terms: Public domain W3C validator