Theorem uniexb 4614

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 4580	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		pwuni 4324	. . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
3		pwexg 4312	. . 3 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V)
4		ssexg 4267	. . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
5	2, 3, 4	sylancr 418	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
6	1, 5	impbii 126	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 105   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3685  ∪ cuni 3930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-uni 3931

This theorem is referenced by:  pwexb  4615  elpwpwel  4616  tfrlemibex  6590  tfr1onlembex  6606  tfrcllembex  6619  ixpexgg  6994  ptex  13595  tgss2  15103  txbasex  15281