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Theorem uniexb 4394
 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 4361 . 2
2 pwuni 4116 . . 3
3 pwexg 4104 . . 3
4 ssexg 4067 . . 3
52, 3, 4sylancr 410 . 2
61, 5impbii 125 1
 Colors of variables: wff set class Syntax hints:   wb 104   wcel 1480  cvv 2686   wss 3071  cpw 3510  cuni 3736 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-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355 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-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737 This theorem is referenced by:  pwexb  4395  elpwpwel  4396  tfrlemibex  6226  tfr1onlembex  6242  tfrcllembex  6255  ixpexgg  6616  tgss2  12248  txbasex  12426
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