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Mirrors > Home > MPE Home > Th. List > prfi | Structured version Visualization version GIF version |
Description: An unordered pair is finite. (Contributed by NM, 22-Aug-2008.) |
Ref | Expression |
---|---|
prfi | ⊢ {𝐴, 𝐵} ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4128 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snfi 7923 | . . 3 ⊢ {𝐴} ∈ Fin | |
3 | snfi 7923 | . . 3 ⊢ {𝐵} ∈ Fin | |
4 | unfi 8112 | . . 3 ⊢ (({𝐴} ∈ Fin ∧ {𝐵} ∈ Fin) → ({𝐴} ∪ {𝐵}) ∈ Fin) | |
5 | 2, 3, 4 | mp2an 704 | . 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ Fin |
6 | 1, 5 | eqeltri 2684 | 1 ⊢ {𝐴, 𝐵} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ∪ cun 3538 {csn 4125 {cpr 4127 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 |
This theorem is referenced by: tpfi 8121 fiint 8122 inelfi 8207 tskpr 9471 hashpw 13083 hashfun 13084 pr2pwpr 13116 hashtpg 13121 sumpr 14321 lcmfpr 15178 isprm2lem 15232 prmreclem2 15459 acsfn2 16147 isdrs2 16762 symg2hash 17640 psgnprfval 17764 znidomb 19729 m2detleib 20256 ovolioo 23143 i1f1 23263 itgioo 23388 limcun 23465 aannenlem2 23888 wilthlem2 24595 perfectlem2 24755 upgrex 25759 umgraex 25852 konigsberg 26514 ex-hash 26702 inelpisys 29544 coinfliplem 29867 coinflippv 29872 subfacp1lem1 30415 poimirlem9 32588 kelac2lem 36652 sumpair 38217 refsum2cnlem1 38219 ibliooicc 38863 fourierdlem50 39049 fourierdlem51 39050 fourierdlem54 39053 fourierdlem70 39069 fourierdlem71 39070 fourierdlem76 39075 fourierdlem102 39101 fourierdlem103 39102 fourierdlem104 39103 fourierdlem114 39113 saluncl 39213 sge0pr 39287 meadjun 39355 omeunle 39406 perfectALTVlem2 40165 zlmodzxzel 41926 gsumpr 41932 ldepspr 42056 zlmodzxzldeplem2 42084 |
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