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Mirrors > Home > NFE Home > Th. List > ncfindi | Unicode version |
Description: Distribution law for finite cardinality. (Contributed by SF, 30-Jan-2015.) |
Ref | Expression |
---|---|
ncfindi | Fin Ncfin Ncfin Ncfin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 979 | . . . 4 Fin Fin | |
2 | simp1r 980 | . . . . 5 Fin | |
3 | simp2 956 | . . . . 5 Fin | |
4 | unexg 4101 | . . . . 5 | |
5 | 2, 3, 4 | syl2anc 642 | . . . 4 Fin |
6 | ncfinprop 4474 | . . . 4 Fin Ncfin Nn Ncfin | |
7 | 1, 5, 6 | syl2anc 642 | . . 3 Fin Ncfin Nn Ncfin |
8 | 7 | simpld 445 | . 2 Fin Ncfin Nn |
9 | ncfinprop 4474 | . . . . 5 Fin Ncfin Nn Ncfin | |
10 | 1, 2, 9 | syl2anc 642 | . . . 4 Fin Ncfin Nn Ncfin |
11 | 10 | simpld 445 | . . 3 Fin Ncfin Nn |
12 | ncfinprop 4474 | . . . . 5 Fin Ncfin Nn Ncfin | |
13 | 1, 3, 12 | syl2anc 642 | . . . 4 Fin Ncfin Nn Ncfin |
14 | 13 | simpld 445 | . . 3 Fin Ncfin Nn |
15 | nncaddccl 4419 | . . 3 Ncfin Nn Ncfin Nn Ncfin Ncfin Nn | |
16 | 11, 14, 15 | syl2anc 642 | . 2 Fin Ncfin Ncfin Nn |
17 | 7 | simprd 449 | . 2 Fin Ncfin |
18 | 10 | simprd 449 | . . 3 Fin Ncfin |
19 | 13 | simprd 449 | . . 3 Fin Ncfin |
20 | simp3 957 | . . 3 Fin | |
21 | eladdci 4399 | . . 3 Ncfin Ncfin Ncfin Ncfin | |
22 | 18, 19, 20, 21 | syl3anc 1182 | . 2 Fin Ncfin Ncfin |
23 | nnceleq 4430 | . 2 Ncfin Nn Ncfin Ncfin Nn Ncfin Ncfin Ncfin Ncfin Ncfin Ncfin | |
24 | 8, 16, 17, 22, 23 | syl22anc 1183 | 1 Fin Ncfin Ncfin Ncfin |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 w3a 934 wceq 1642 wcel 1710 cvv 2859 cun 3207 cin 3208 c0 3550 Nn cnnc 4373 cplc 4375 Fin cfin 4376 Ncfin cncfin 4434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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: vfintle 4546 vfin1cltv 4547 vfinncsp 4554 |
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