Theorem uniexb 4520

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 4486	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		pwuni 4236	. . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
3		pwexg 4224	. . 3 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V)
4		ssexg 4183	. . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
5	2, 3, 4	sylancr 414	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
6	1, 5	impbii 126	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 105   ∈ wcel 2176  Vcvv 2772   ⊆ wss 3166  𝒫 cpw 3616  ∪ cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-uni 3851

This theorem is referenced by:  pwexb  4521  elpwpwel  4522  tfrlemibex  6415  tfr1onlembex  6431  tfrcllembex  6444  ixpexgg  6809  ptex  13096  tgss2  14551  txbasex  14729