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	⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Proof of Theorem uniexb

Step	Hyp	Ref	Expression
1		uniexg 4361	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		pwuni 4116	. . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
3		pwexg 4104	. . 3 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V)
4		ssexg 4067	. . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
5	2, 3, 4	sylancr 410	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
6	1, 5	impbii 125	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 104   ∈ wcel 1480  Vcvv 2686   ⊆ wss 3071  𝒫 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