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Mirrors > Home > ILE Home > Th. List > uniexb | GIF version |
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.) |
Ref | Expression |
---|---|
uniexb | ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 4438 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
2 | pwuni 4191 | . . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
3 | pwexg 4179 | . . 3 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V) | |
4 | ssexg 4141 | . . 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) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 𝒫 cpw 3575 ∪ cuni 3809 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-un 4432 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 df-uni 3810 |
This theorem is referenced by: pwexb 4473 elpwpwel 4474 tfrlemibex 6327 tfr1onlembex 6343 tfrcllembex 6356 ixpexgg 6719 tgss2 13450 txbasex 13628 |
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