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Mirrors > Home > NFE Home > Th. List > ncssfin | Unicode version |
Description: A cardinal is finite iff it is a subset of Fin. (Contributed by SF, 25-Feb-2015.) |
Ref | Expression |
---|---|
ncssfin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 3919 |
. . 3
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2 | df-fin 4380 |
. . 3
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3 | 1, 2 | syl6sseqr 3318 |
. 2
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4 | dfss2 3262 |
. . . 4
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5 | elfin 4420 |
. . . . . . 7
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6 | 5 | imbi2i 303 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | peano1 4402 |
. . . . . . 7
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8 | ne0i 3556 |
. . . . . . 7
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9 | r19.37zv 3646 |
. . . . . . 7
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10 | 7, 8, 9 | mp2b 9 |
. . . . . 6
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11 | 6, 10 | bitr4i 243 |
. . . . 5
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12 | 11 | albii 1566 |
. . . 4
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13 | 4, 12 | bitri 240 |
. . 3
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14 | nulnnc 6118 |
. . . . . . 7
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15 | eleq1 2413 |
. . . . . . 7
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16 | 14, 15 | mtbiri 294 |
. . . . . 6
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17 | 16 | necon2ai 2561 |
. . . . 5
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18 | n0 3559 |
. . . . 5
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19 | 17, 18 | sylib 188 |
. . . 4
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20 | 19.29r 1597 |
. . . . 5
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21 | pm2.27 35 |
. . . . . . . . . . 11
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22 | 21 | adantl 452 |
. . . . . . . . . 10
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23 | nnnc 6146 |
. . . . . . . . . . . . 13
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24 | nceleq 6149 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | sylanl2 632 |
. . . . . . . . . . . 12
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26 | simplr 731 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | eqeltrd 2427 |
. . . . . . . . . . 11
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28 | 27 | expr 598 |
. . . . . . . . . 10
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29 | 22, 28 | syld 40 |
. . . . . . . . 9
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30 | 29 | an32s 779 |
. . . . . . . 8
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31 | 30 | rexlimdva 2738 |
. . . . . . 7
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32 | 31 | expimpd 586 |
. . . . . 6
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33 | 32 | exlimdv 1636 |
. . . . 5
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34 | 20, 33 | syl5 28 |
. . . 4
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35 | 19, 34 | mpand 656 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 13, 35 | syl5bi 208 |
. 2
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37 | 3, 36 | impbid2 195 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-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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