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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 4574 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prfi 8795 | . . 3 ⊢ {𝐴, 𝐵} ∈ Fin | |
3 | snfi 8596 | . . 3 ⊢ {𝐶} ∈ Fin | |
4 | unfi 8787 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
5 | 2, 3, 4 | mp2an 690 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin |
6 | 1, 5 | eqeltri 2911 | 1 ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ∪ cun 3936 {csn 4569 {cpr 4571 {ctp 4573 Fincfn 8511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-fin 8515 |
This theorem is referenced by: hashge3el3dif 13847 sumtp 15106 lcmftp 15982 perfectlem2 25808 prodtp 30545 hgt750lemg 31927 limsupequzlem 42010 fourierdlem102 42500 fourierdlem114 42512 etransclem48 42574 perfectALTVlem2 43894 |
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