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Mirrors > Home > MPE Home > Th. List > tpfi | Structured version Visualization version GIF version |
Description: An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.) |
Ref | Expression |
---|---|
tpfi | ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4632 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prfi 9324 | . . 3 ⊢ {𝐴, 𝐵} ∈ Fin | |
3 | snfi 9046 | . . 3 ⊢ {𝐶} ∈ Fin | |
4 | unfi 9174 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
5 | 2, 3, 4 | mp2an 688 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin |
6 | 1, 5 | eqeltri 2827 | 1 ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2104 ∪ cun 3945 {csn 4627 {cpr 4629 {ctp 4631 Fincfn 8941 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7858 df-1o 8468 df-en 8942 df-fin 8945 |
This theorem is referenced by: hashge3el3dif 14451 sumtp 15699 lcmftp 16577 perfectlem2 26969 prodtp 32300 hgt750lemg 33964 limsupequzlem 44736 fourierdlem102 45222 fourierdlem114 45234 etransclem48 45296 perfectALTVlem2 46688 |
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