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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uniexd | Structured version Visualization version GIF version |
Description: Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
uniexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
uniexd | ⊢ (𝜑 → ∪ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | uniexg 7189 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∪ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 Vcvv 3385 ∪ cuni 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 ax-sep 4975 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rex 3095 df-v 3387 df-uni 4629 |
This theorem is referenced by: restuni4 40062 subsaluni 41321 issmflem 41682 issmflelem 41699 issmfle 41700 smfconst 41704 issmfgtlem 41710 issmfgt 41711 issmfgelem 41723 issmfge 41724 smfpimioo 41740 smfresal 41741 |
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