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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 7677. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
uniexr | ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4911 | . 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | pwexg 5338 | . 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V) | |
3 | ssexg 5285 | . 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V) | |
4 | 1, 2, 3 | sylancr 588 | 1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3448 ⊆ wss 3915 𝒫 cpw 4565 ∪ cuni 4870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-pow 5325 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-in 3922 df-ss 3932 df-pw 4567 df-uni 4871 |
This theorem is referenced by: uniexb 7703 ssonprc 7727 ac5num 9979 bj-restv 35595 bj-mooreset 35602 ipoglb0 47093 |
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