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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 4633 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snfi 9069 | . . 3 ⊢ {𝐴} ∈ Fin | |
3 | snfi 9069 | . . 3 ⊢ {𝐵} ∈ Fin | |
4 | unfi 9197 | . . 3 ⊢ (({𝐴} ∈ Fin ∧ {𝐵} ∈ Fin) → ({𝐴} ∪ {𝐵}) ∈ Fin) | |
5 | 2, 3, 4 | mp2an 690 | . 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ Fin |
6 | 1, 5 | eqeltri 2821 | 1 ⊢ {𝐴, 𝐵} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∪ cun 3942 {csn 4630 {cpr 4632 Fincfn 8964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-om 7872 df-1o 8487 df-en 8965 df-fin 8968 |
This theorem is referenced by: tpfi 9349 fiint 9350 inelfi 9443 tskpr 10795 hashpw 14431 hashfun 14432 pr2pwpr 14476 hashtpg 14482 sumpr 15730 lcmfpr 16601 prmreclem2 16889 acsfn2 17646 isdrs2 18301 efmnd2hash 18854 symg2hash 19358 psgnprfval 19488 gsumpr 19922 znidomb 21512 m2detleib 22577 ovolioo 25541 i1f1 25663 itgioo 25789 limcun 25868 aannenlem2 26309 wilthlem2 27046 perfectlem2 27208 upgrex 28977 ex-hash 30335 prodpr 32674 linds2eq 33193 elrspunsn 33241 inelpisys 33904 coinfliplem 34229 coinflippv 34234 subfacp1lem1 34920 poimirlem9 37233 kelac2lem 42630 sumpair 44539 refsum2cnlem1 44541 climxlim2lem 45371 ibliooicc 45497 fourierdlem50 45682 fourierdlem51 45683 fourierdlem54 45686 fourierdlem70 45702 fourierdlem71 45703 fourierdlem76 45708 fourierdlem102 45734 fourierdlem103 45735 fourierdlem104 45736 fourierdlem114 45746 saluncl 45843 sge0pr 45920 meadjun 45988 omeunle 46042 perfectALTVlem2 47199 zlmodzxzel 47605 ldepspr 47727 zlmodzxzldeplem2 47755 rrx2line 47999 2sphere 48008 |
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