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Mirrors > Home > NFE Home > Th. List > snfi | Unicode version |
Description: A singleton is finite. (Contributed by SF, 23-Feb-2015.) |
Ref | Expression |
---|---|
snfi | Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnnc 4408 | . . . 4 1c Nn | |
2 | snel1cg 4141 | . . . 4 1c | |
3 | eleq2 2414 | . . . . 5 1c 1c | |
4 | 3 | rspcev 2955 | . . . 4 1c Nn 1c Nn |
5 | 1, 2, 4 | sylancr 644 | . . 3 Nn |
6 | elfin 4420 | . . 3 Fin Nn | |
7 | 5, 6 | sylibr 203 | . 2 Fin |
8 | snprc 3788 | . . 3 | |
9 | 0fin 4423 | . . . 4 Fin | |
10 | eleq1 2413 | . . . 4 Fin Fin | |
11 | 9, 10 | mpbiri 224 | . . 3 Fin |
12 | 8, 11 | sylbi 187 | . 2 Fin |
13 | 7, 12 | pm2.61i 156 | 1 Fin |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wceq 1642 wcel 1710 wrex 2615 cvv 2859 c0 3550 csn 3737 1cc1c 4134 Nn cnnc 4373 Fin cfin 4376 |
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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-nul 3551 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-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-p6 4191 df-sik 4192 df-ssetk 4193 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 |
This theorem is referenced by: nchoicelem19 6307 |
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