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Mirrors > Home > NFE Home > Th. List > ncfinprop | Unicode version |
Description: Properties of finite cardinal number. Theorem X.1.23 of [Rosser] p. 527 (Contributed by SF, 20-Jan-2015.) |
Ref | Expression |
---|---|
ncfinprop | Fin Ncfin Nn Ncfin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ncfin 4442 | . . . 4 Ncfin Nn | |
2 | vfinnc 4471 | . . . . 5 Fin Nn | |
3 | reiotacl 4364 | . . . . 5 Nn Nn Nn | |
4 | 2, 3 | syl 15 | . . . 4 Fin Nn Nn |
5 | 1, 4 | syl5eqel 2437 | . . 3 Fin Ncfin Nn |
6 | 1 | eqcomi 2357 | . . . 4 Nn Ncfin |
7 | eleq2 2414 | . . . . . 6 Ncfin Ncfin | |
8 | 7 | reiota2 4368 | . . . . 5 Ncfin Nn Nn Ncfin Nn Ncfin |
9 | 5, 2, 8 | syl2anc 642 | . . . 4 Fin Ncfin Nn Ncfin |
10 | 6, 9 | mpbiri 224 | . . 3 Fin Ncfin |
11 | 5, 10 | jca 518 | . 2 Fin Ncfin Nn Ncfin |
12 | 11 | ancoms 439 | 1 Fin Ncfin Nn Ncfin |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wceq 1642 wcel 1710 wreu 2616 cvv 2859 cio 4337 Nn cnnc 4373 Fin cfin 4376 Ncfin cncfin 4434 |
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-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-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-ncfin 4442 |
This theorem is referenced by: ncfindi 4475 ncfinsn 4476 ncfineleq 4477 ncfintfin 4495 vfinspnn 4541 1cvsfin 4542 vfintle 4546 vfin1cltv 4547 vfinncvntnn 4548 vfinspsslem1 4550 vfinncsp 4554 vinf 4555 |
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