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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 4324 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snfi 8203 | . . 3 ⊢ {𝐴} ∈ Fin | |
3 | snfi 8203 | . . 3 ⊢ {𝐵} ∈ Fin | |
4 | unfi 8392 | . . 3 ⊢ (({𝐴} ∈ Fin ∧ {𝐵} ∈ Fin) → ({𝐴} ∪ {𝐵}) ∈ Fin) | |
5 | 2, 3, 4 | mp2an 710 | . 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ Fin |
6 | 1, 5 | eqeltri 2835 | 1 ⊢ {𝐴, 𝐵} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 ∪ cun 3713 {csn 4321 {cpr 4323 Fincfn 8121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-fin 8125 |
This theorem is referenced by: tpfi 8401 fiint 8402 inelfi 8489 tskpr 9784 hashpw 13415 hashfun 13416 pr2pwpr 13453 hashtpg 13459 sumpr 14676 lcmfpr 15542 prmreclem2 15823 acsfn2 16525 isdrs2 17140 symg2hash 18017 psgnprfval 18141 znidomb 20112 m2detleib 20639 ovolioo 23536 i1f1 23656 itgioo 23781 limcun 23858 aannenlem2 24283 wilthlem2 24994 perfectlem2 25154 upgrex 26186 ex-hash 27621 prodpr 29881 inelpisys 30526 coinfliplem 30849 coinflippv 30854 subfacp1lem1 31468 poimirlem9 33731 kelac2lem 38136 sumpair 39693 refsum2cnlem1 39695 climxlim2lem 40574 ibliooicc 40690 fourierdlem50 40876 fourierdlem51 40877 fourierdlem54 40880 fourierdlem70 40896 fourierdlem71 40897 fourierdlem76 40902 fourierdlem102 40928 fourierdlem103 40929 fourierdlem104 40930 fourierdlem114 40940 saluncl 41040 sge0pr 41114 meadjun 41182 omeunle 41236 perfectALTVlem2 42141 zlmodzxzel 42643 gsumpr 42649 ldepspr 42772 zlmodzxzldeplem2 42800 |
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