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Mirrors > Home > NFE Home > Th. List > ltfinirr | Unicode version |
Description: Irreflexive law for finite less than. (Contributed by SF, 29-Jan-2015.) |
Ref | Expression |
---|---|
ltfinirr | Nn fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cnsuc 4401 | . . . . . . . 8 1c 0c | |
2 | 1 | necomi 2598 | . . . . . . 7 0c 1c |
3 | df-ne 2518 | . . . . . . 7 0c 1c 0c 1c | |
4 | 2, 3 | mpbi 199 | . . . . . 6 0c 1c |
5 | addcid1 4405 | . . . . . . . . 9 0c | |
6 | 5 | eqcomi 2357 | . . . . . . . 8 0c |
7 | addcass 4415 | . . . . . . . 8 1c 1c | |
8 | 6, 7 | eqeq12i 2366 | . . . . . . 7 1c 0c 1c |
9 | simpll 730 | . . . . . . . 8 Nn Nn Nn | |
10 | peano1 4402 | . . . . . . . . 9 0c Nn | |
11 | 10 | a1i 10 | . . . . . . . 8 Nn Nn 0c Nn |
12 | peano2 4403 | . . . . . . . . 9 Nn 1c Nn | |
13 | 12 | adantl 452 | . . . . . . . 8 Nn Nn 1c Nn |
14 | 5 | neeq1i 2526 | . . . . . . . . . 10 0c |
15 | 14 | biimpri 197 | . . . . . . . . 9 0c |
16 | 15 | ad2antlr 707 | . . . . . . . 8 Nn Nn 0c |
17 | preaddccan2 4455 | . . . . . . . 8 Nn 0c Nn 1c Nn 0c 0c 1c 0c 1c | |
18 | 9, 11, 13, 16, 17 | syl31anc 1185 | . . . . . . 7 Nn Nn 0c 1c 0c 1c |
19 | 8, 18 | syl5bb 248 | . . . . . 6 Nn Nn 1c 0c 1c |
20 | 4, 19 | mtbiri 294 | . . . . 5 Nn Nn 1c |
21 | 20 | nrexdv 2717 | . . . 4 Nn Nn 1c |
22 | 21 | ex 423 | . . 3 Nn Nn 1c |
23 | imnan 411 | . . 3 Nn 1c Nn 1c | |
24 | 22, 23 | sylib 188 | . 2 Nn Nn 1c |
25 | opkltfing 4449 | . . 3 Nn Nn fin Nn 1c | |
26 | 25 | anidms 626 | . 2 Nn fin Nn 1c |
27 | 24, 26 | mtbird 292 | 1 Nn fin |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wceq 1642 wcel 1710 wne 2516 wrex 2615 c0 3550 copk 4057 1cc1c 4134 Nn cnnc 4373 0cc0c 4374 cplc 4375 fin cltfin 4433 |
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-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-0c 4377 df-addc 4378 df-nnc 4379 df-ltfin 4441 |
This theorem is referenced by: ltfinasym 4460 lenltfin 4469 tfinltfin 4501 sfin111 4536 vfinncvntnn 4548 |
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