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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 7319. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
uniexr | ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4781 | . 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | pwexg 5170 | . 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V) | |
3 | ssexg 5118 | . 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V) | |
4 | 1, 2, 3 | sylancr 587 | 1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3859 𝒫 cpw 4453 ∪ cuni 4745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-pow 5157 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-in 3866 df-ss 3874 df-pw 4455 df-uni 4746 |
This theorem is referenced by: uniexb 7343 ssonprc 7363 ac5num 9308 bj-restv 33985 bj-mooreset 33993 |
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