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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 9363 | . . 3 ⊢ {𝐴, 𝐵} ∈ Fin | |
| 3 | snfi 9083 | . . 3 ⊢ {𝐶} ∈ Fin | |
| 4 | unfi 9211 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
| 5 | 2, 3, 4 | mp2an 692 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin |
| 6 | 1, 5 | eqeltri 2837 | 1 ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 Fincfn 8985 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-2o 8507 df-en 8986 df-fin 8989 |
| This theorem is referenced by: hash7g 14525 hashge3el3dif 14526 tpf1o 14540 s7f1o 15005 sumtp 15785 lcmftp 16673 perfectlem2 27274 prodtp 32829 gsumtp 33061 hgt750lemg 34669 limsupequzlem 45737 fourierdlem102 46223 fourierdlem114 46235 etransclem48 46297 perfectALTVlem2 47709 |
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