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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 4923 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {cpr 4628 ∪ cuni 4907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 df-uni 4908 |
| This theorem is referenced by: uniintsn 4985 uniop 5520 unexOLD 7765 nlim2 8528 rankxplim 9919 mrcun 17665 indistps 23018 indistps2 23019 leordtval2 23220 ex-uni 30445 mnuprdlem1 44291 mnuprdlem2 44292 mnurndlem1 44300 fouriersw 46246 |
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