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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 4924 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∪ cun 3945 {cpr 4631 ∪ cuni 4908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-un 3952 df-sn 4630 df-pr 4632 df-uni 4909 |
This theorem is referenced by: uniprgOLD 4927 uniintsn 4990 uniop 5517 unex 7748 nlim2 8510 rankxplim 9902 mrcun 17601 indistps 22913 indistps2 22914 leordtval2 23115 ex-uni 30235 mnuprdlem1 43709 mnuprdlem2 43710 mnurndlem1 43718 fouriersw 45619 |
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