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| Mirrors > Home > MPE Home > Th. List > xpfi | Structured version Visualization version GIF version | ||
| Description: The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 10-Jan-2025.) |
| Ref | Expression |
|---|---|
| xpfi | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unfi 9211 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
| 2 | pwfi 9357 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin ↔ 𝒫 (𝐴 ∪ 𝐵) ∈ Fin) | |
| 3 | pwfi 9357 | . . . 4 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∈ Fin ↔ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ Fin) | |
| 4 | 2, 3 | bitri 275 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin ↔ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ Fin) |
| 5 | 1, 4 | sylib 218 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ Fin) |
| 6 | xpsspw 5819 | . 2 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
| 7 | ssfi 9213 | . 2 ⊢ ((𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) → (𝐴 × 𝐵) ∈ Fin) | |
| 8 | 5, 6, 7 | sylancl 586 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 × cxp 5683 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-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 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-en 8986 df-dom 8987 df-fin 8989 |
| This theorem is referenced by: 3xpfi 9360 fodomfir 9368 mapfi 9388 fsuppxpfi 9425 infxpenlem 10053 ficardadju 10240 ackbij1lem9 10267 ackbij1lem10 10268 hashxplem 14472 hashmap 14474 fsum2dlem 15806 fsumcom2 15810 ackbijnn 15864 fprod2dlem 16016 fprodcom2 16020 rexpen 16264 crth 16815 phimullem 16816 prmreclem3 16956 gsumcom3fi 19997 ablfaclem3 20107 gsumdixp 20316 frlmbas3 21796 gsumbagdiag 21951 psrass1lem 21952 evlslem2 22103 mamudm 22399 mamufacex 22400 mamures 22401 mamucl 22405 mamudi 22407 mamudir 22408 mamuvs1 22409 mamuvs2 22410 matsca2 22426 matbas2 22427 matplusg2 22433 matvsca2 22434 matplusgcell 22439 matsubgcell 22440 matvscacell 22442 matgsum 22443 mamumat1cl 22445 mattposcl 22459 mdetrsca 22609 mdetunilem9 22626 pmatcoe1fsupp 22707 tsmsxplem1 24161 tsmsxplem2 24162 tsmsxp 24163 i1fadd 25730 i1fmul 25731 itg1addlem4 25734 fsumdvdsmul 27238 fsumdvdsmulOLD 27240 fsumvma 27257 lgsquadlem1 27424 lgsquadlem2 27425 lgsquadlem3 27426 madefi 27950 relfi 32615 fsumiunle 32831 elrgspnlem2 33247 matdim 33666 fedgmullem1 33680 fldextrspunlsplem 33723 sibfof 34342 hgt750lemb 34671 erdszelem10 35205 matunitlindflem2 37624 matunitlindf 37625 poimirlem26 37653 poimirlem27 37654 poimirlem28 37655 cntotbnd 37803 aks6d1c2 42131 sticksstones22 42169 pellex 42846 mnringmulrcld 44247 fourierdlem42 46164 etransclem44 46293 etransclem45 46294 etransclem47 46296 |
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