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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 4631 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prfi 9317 | . . 3 ⊢ {𝐴, 𝐵} ∈ Fin | |
3 | snfi 9039 | . . 3 ⊢ {𝐶} ∈ Fin | |
4 | unfi 9167 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
5 | 2, 3, 4 | mp2an 691 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin |
6 | 1, 5 | eqeltri 2830 | 1 ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ∪ cun 3944 {csn 4626 {cpr 4628 {ctp 4630 Fincfn 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-om 7850 df-1o 8460 df-en 8935 df-fin 8938 |
This theorem is referenced by: hashge3el3dif 14442 sumtp 15690 lcmftp 16568 perfectlem2 26712 prodtp 32010 hgt750lemg 33603 limsupequzlem 44372 fourierdlem102 44858 fourierdlem114 44870 etransclem48 44932 perfectALTVlem2 46324 |
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