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Mirrors > Home > MPE Home > Th. List > 0fin | Structured version Visualization version GIF version |
Description: The empty set is finite. (Contributed by FL, 14-Jul-2008.) |
Ref | Expression |
---|---|
0fin | ⊢ ∅ ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 6977 | . 2 ⊢ ∅ ∈ ω | |
2 | ssid 3587 | . 2 ⊢ ∅ ⊆ ∅ | |
3 | ssnnfi 8064 | . 2 ⊢ ((∅ ∈ ω ∧ ∅ ⊆ ∅) → ∅ ∈ Fin) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ ∅ ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 ωcom 6957 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-rab 2905 df-v 3175 df-sbc 3403 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-br 4584 df-opab 4644 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-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-om 6958 df-en 7842 df-fin 7845 |
This theorem is referenced by: nfielex 8074 xpfi 8116 fnfi 8123 iunfi 8137 fczfsuppd 8176 fsuppun 8177 0fsupp 8180 r1fin 8519 acndom 8757 numwdom 8765 ackbij1lem18 8942 sdom2en01 9007 fin23lem26 9030 isfin1-3 9091 gchxpidm 9370 fzfi 12633 fzofi 12635 hasheq0 13015 hashxp 13081 lcmf0 15185 0hashbc 15549 acsfn0 16144 isdrs2 16762 fpwipodrs 16987 symgfisg 17711 mplsubg 19258 mpllss 19259 psrbag0 19315 dsmm0cl 19903 mat0dimbas0 20091 mat0dim0 20092 mat0dimid 20093 mat0dimscm 20094 mat0dimcrng 20095 mat0scmat 20163 mavmul0 20177 mavmul0g 20178 mdet0pr 20217 m1detdiag 20222 d0mat2pmat 20362 chpmat0d 20458 fctop 20618 cmpfi 21021 bwth 21023 comppfsc 21145 ptbasid 21188 cfinfil 21507 ufinffr 21543 fin1aufil 21546 alexsubALTlem2 21662 alexsubALTlem4 21664 ptcmplem2 21667 tsmsfbas 21741 xrge0gsumle 22444 xrge0tsms 22445 fta1 23867 wwlknfi 26266 xrge0tsmsd 29116 esumnul 29437 esum0 29438 esumcst 29452 esumsnf 29453 esumpcvgval 29467 sibf0 29723 eulerpartlemt 29760 derang0 30405 topdifinffinlem 32371 matunitlindf 32577 0totbnd 32742 heiborlem6 32785 mzpcompact2lem 36332 rp-isfinite6 36883 0pwfi 38252 fouriercn 39125 rrxtopn0 39189 salexct 39228 sge0rnn0 39261 sge00 39269 sge0sn 39272 ovn0val 39440 ovn02 39458 hoidmv0val 39473 hoidmvle 39490 hoiqssbl 39515 von0val 39562 vonhoire 39563 vonioo 39573 vonicc 39576 vonsn 39582 uhgr0edgfi 40466 fusgrfisbase 40547 vtxdg0e 40689 wwlksnfi 41112 wwlksnonfi 41127 clwwlksnfi 41220 av-numclwwlkffin0 41513 lcoc0 42005 lco0 42010 |
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