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Mirrors > Home > NFE Home > Th. List > 0lt1c | Unicode version |
Description: Cardinal one is strictly greater than cardinal zero. (Contributed by Scott Fenton, 1-Aug-2019.) |
Ref | Expression |
---|---|
0lt1c | 0c c 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df0c2 6137 | . . . 4 0c Nc | |
2 | 0ss 3579 | . . . . 5 | |
3 | 0ex 4110 | . . . . . 6 | |
4 | snex 4111 | . . . . . 6 | |
5 | 3, 4 | nclec 6195 | . . . . 5 Nc c Nc |
6 | 2, 5 | ax-mp 5 | . . . 4 Nc c Nc |
7 | 1, 6 | eqbrtri 4658 | . . 3 0c c Nc |
8 | vex 2862 | . . . . . . 7 | |
9 | 8 | snnz 3834 | . . . . . 6 |
10 | df-ne 2518 | . . . . . 6 | |
11 | 9, 10 | mpbi 199 | . . . . 5 |
12 | 4 | ncid 6123 | . . . . . . 7 Nc |
13 | eleq2 2414 | . . . . . . 7 0c Nc 0c Nc | |
14 | 12, 13 | mpbiri 224 | . . . . . 6 0c Nc 0c |
15 | el0c 4421 | . . . . . 6 0c | |
16 | 14, 15 | sylib 188 | . . . . 5 0c Nc |
17 | 11, 16 | mto 167 | . . . 4 0c Nc |
18 | df-ne 2518 | . . . 4 0c Nc 0c Nc | |
19 | 17, 18 | mpbir 200 | . . 3 0c Nc |
20 | brltc 6114 | . . 3 0c c Nc 0c c Nc 0c Nc | |
21 | 7, 19, 20 | mpbir2an 886 | . 2 0c c Nc |
22 | 8 | df1c3 6140 | . 2 1c Nc |
23 | 21, 22 | breqtrri 4664 | 1 0c c 1c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wceq 1642 wcel 1710 wne 2516 wss 3257 c0 3550 csn 3737 1cc1c 4134 0cc0c 4374 class class class wbr 4639 c clec 6089 c cltc 6090 Nc cnc 6091 |
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-lec 6099 df-ltc 6100 df-nc 6101 |
This theorem is referenced by: nmembers1lem2 6269 |
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