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Theorem uniexb 4171
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 (A V ↔ A V)

Proof of Theorem uniexb
StepHypRef Expression
1 uniexg 4141 . 2 (A V → A V)
2 pwuni 3934 . . 3 A ⊆ 𝒫 A
3 pwexg 3924 . . 3 ( A V → 𝒫 A V)
4 ssexg 3887 . . 3 ((A ⊆ 𝒫 A 𝒫 A V) → A V)
52, 3, 4sylancr 393 . 2 ( A V → A V)
61, 5impbii 117 1 (A V ↔ A V)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1390  Vcvv 2551  wss 2911  𝒫 cpw 3351   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572
This theorem is referenced by:  pwexb  4172  tfrlemibex  5884
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