Theorem uniexb 4205

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 4175	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		pwuni 3943	. . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
3		pwexg 3933	. . 3 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V)
4		ssexg 3896	. . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
5	2, 3, 4	sylancr 393	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
6	1, 5	impbii 117	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 98   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359  ∪ cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581

This theorem is referenced by:  pwexb  4206  tfrlemibex  5943