![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 0fin | GIF version |
Description: The empty set is finite. (Contributed by FL, 14-Jul-2008.) |
Ref | Expression |
---|---|
0fin | ⊢ ∅ ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 4468 | . 2 ⊢ ∅ ∈ ω | |
2 | nnfi 6719 | . 2 ⊢ (∅ ∈ ω → ∅ ∈ Fin) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ ∅ ∈ Fin |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1463 ∅c0 3329 ωcom 4464 Fincfn 6588 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-id 4175 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-en 6589 df-fin 6591 |
This theorem is referenced by: xpfi 6771 ssfirab 6774 fnfi 6777 iunfidisj 6786 fidcenumlemr 6795 fzfig 10096 fihasheq0 10433 hash0 10436 |
Copyright terms: Public domain | W3C validator |