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Mirrors > Home > MPE Home > Th. List > unipr | Structured version Visualization version GIF version |
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) (Proof shortened by BJ, 1-Sep-2024.) |
Ref | Expression |
---|---|
unipr.1 | ⊢ 𝐴 ∈ V |
unipr.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
unipr | ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unipr.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | unipr.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | uniprg 4916 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∪ cun 3939 {cpr 4623 ∪ cuni 4900 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-un 3946 df-sn 4622 df-pr 4624 df-uni 4901 |
This theorem is referenced by: uniprgOLD 4919 uniintsn 4982 uniop 5506 unex 7727 nlim2 8486 rankxplim 9871 mrcun 17571 indistps 22858 indistps2 22859 leordtval2 23060 ex-uni 30173 mnuprdlem1 43580 mnuprdlem2 43581 mnurndlem1 43589 fouriersw 45492 |
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