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

Theorem uniexb 4205
 Description: The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexb

Proof of Theorem uniexb
StepHypRef Expression
1 uniexg 4175 . 2
2 pwuni 3943 . . 3
3 pwexg 3933 . . 3
4 ssexg 3896 . . 3
52, 3, 4sylancr 393 . 2
61, 5impbii 117 1
 Colors of variables: wff set class Syntax hints:   wb 98   wcel 1393  cvv 2557   wss 2917  cpw 3359  cuni 3580 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581 This theorem is referenced by:  pwexb  4206  tfrlemibex  5943
 Copyright terms: Public domain W3C validator