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Theorem uniexb 4403
 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 4370 . 2 (𝐴 ∈ V → 𝐴 ∈ V)
2 pwuni 4125 . . 3 𝐴 ⊆ 𝒫 𝐴
3 pwexg 4113 . . 3 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ V)
4 ssexg 4076 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
52, 3, 4sylancr 411 . 2 ( 𝐴 ∈ V → 𝐴 ∈ V)
61, 5impbii 125 1 (𝐴 ∈ V ↔ 𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∈ wcel 1481  Vcvv 2690   ⊆ wss 3077  𝒫 cpw 3516  ∪ cuni 3745 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2692  df-in 3083  df-ss 3090  df-pw 3518  df-uni 3746 This theorem is referenced by:  pwexb  4404  elpwpwel  4405  tfrlemibex  6235  tfr1onlembex  6251  tfrcllembex  6264  ixpexgg  6625  tgss2  12307  txbasex  12485
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