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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 7729. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
uniexr | ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4949 | . 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | pwexg 5376 | . 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V) | |
3 | ssexg 5323 | . 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 2105 Vcvv 3473 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-pow 5363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-in 3955 df-ss 3965 df-pw 4604 df-uni 4909 |
This theorem is referenced by: uniexb 7755 ssonprc 7779 ac5num 10037 bj-restv 36443 bj-mooreset 36450 ipoglb0 47784 |
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