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Theorem uniexb 4232
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 4202 . 2 (𝐴 ∈ V → 𝐴 ∈ V)
2 pwuni 3970 . . 3 𝐴 ⊆ 𝒫 𝐴
3 pwexg 3960 . . 3 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ V)
4 ssexg 3923 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
52, 3, 4sylancr 399 . 2 ( 𝐴 ∈ V → 𝐴 ∈ V)
61, 5impbii 121 1 (𝐴 ∈ V ↔ 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wb 102  wcel 1409  Vcvv 2574  wss 2944  𝒫 cpw 3386   cuni 3607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-uni 3608
This theorem is referenced by:  pwexb  4233  tfrlemibex  5973
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