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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 4836 | . 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 1543 ∈ wcel 2110 Vcvv 3408 ∪ cun 3864 {cpr 4543 ∪ cuni 4819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 df-sn 4542 df-pr 4544 df-uni 4820 |
This theorem is referenced by: uniprgOLD 4839 uniintsn 4898 uniop 5398 unex 7531 rankxplim 9495 mrcun 17125 indistps 21908 indistps2 21909 leordtval2 22109 ex-uni 28509 mnuprdlem1 41563 mnuprdlem2 41564 mnurndlem1 41572 fouriersw 43447 |
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