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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 4411 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
2 | pwuni 4165 | . . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
3 | pwexg 4153 | . . 3 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V) | |
4 | ssexg 4115 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | sylancr 411 | . 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V) |
6 | 1, 5 | impbii 125 | 1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 𝒫 cpw 3553 ∪ cuni 3783 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 df-uni 3784 |
This theorem is referenced by: pwexb 4446 elpwpwel 4447 tfrlemibex 6288 tfr1onlembex 6304 tfrcllembex 6317 ixpexgg 6679 tgss2 12626 txbasex 12804 |
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