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Mirrors > Home > MPE Home > Th. List > uniexr | Structured version Visualization version GIF version |
Description: Converse of the Axiom of Union. Note that it does not require ax-un 7721. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
uniexr | ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4942 | . 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | pwexg 5369 | . 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V) | |
3 | ssexg 5316 | . 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V) | |
4 | 1, 2, 3 | sylancr 586 | 1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 ∪ cuni 4902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-pow 5356 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-in 3950 df-ss 3960 df-pw 4599 df-uni 4903 |
This theorem is referenced by: uniexb 7747 ssonprc 7771 ac5num 10030 bj-restv 36483 bj-mooreset 36490 ipoglb0 47874 |
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