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Mirrors > Home > NFE Home > Th. List > ncfin | Unicode version |
Description: The cardinality of a set is a natural iff the set is finite. (Contributed by SF, 19-Mar-2015.) |
Ref | Expression |
---|---|
ncfin.1 |
Ref | Expression |
---|---|
ncfin | Nc Nn Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ncfin.1 | . . . . 5 | |
2 | 1 | ncid 6123 | . . . 4 Nc |
3 | eleq2 2414 | . . . . 5 Nc Nc | |
4 | 3 | rspcev 2955 | . . . 4 Nc Nn Nc Nn |
5 | 2, 4 | mpan2 652 | . . 3 Nc Nn Nn |
6 | eqcom 2355 | . . . . . . . . 9 Nc Nc | |
7 | nnnc 6146 | . . . . . . . . . 10 Nn NC | |
8 | ncseqnc 6128 | . . . . . . . . . 10 NC Nc | |
9 | 7, 8 | syl 15 | . . . . . . . . 9 Nn Nc |
10 | 6, 9 | syl5bb 248 | . . . . . . . 8 Nn Nc |
11 | 10 | biimpar 471 | . . . . . . 7 Nn Nc |
12 | 11 | eleq1d 2419 | . . . . . 6 Nn Nc Nn Nn |
13 | 12 | exbiri 605 | . . . . 5 Nn Nn Nc Nn |
14 | 13 | pm2.43a 45 | . . . 4 Nn Nc Nn |
15 | 14 | rexlimiv 2732 | . . 3 Nn Nc Nn |
16 | 5, 15 | impbii 180 | . 2 Nc Nn Nn |
17 | elfin 4420 | . 2 Fin Nn | |
18 | 16, 17 | bitr4i 243 | 1 Nc Nn Fin |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 wceq 1642 wcel 1710 wrex 2615 cvv 2859 Nn cnnc 4373 Fin cfin 4376 NC cncs 6088 Nc cnc 6091 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-txp 5736 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-en 6029 df-ncs 6098 df-nc 6101 |
This theorem is referenced by: (None) |
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