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Mirrors > Home > NFE Home > Th. List > ltfinp1 | Unicode version |
Description: One plus a finite cardinal is strictly greater. (Contributed by SF, 29-Jan-2015.) |
Ref | Expression |
---|---|
ltfinp1 | 1c fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 447 | . . 3 | |
2 | peano1 4402 | . . . 4 0c Nn | |
3 | addcid1 4405 | . . . . . 6 0c | |
4 | 3 | addceq1i 4386 | . . . . 5 0c 1c 1c |
5 | 4 | eqcomi 2357 | . . . 4 1c 0c 1c |
6 | addceq2 4384 | . . . . . . 7 0c 0c | |
7 | 6 | addceq1d 4389 | . . . . . 6 0c 1c 0c 1c |
8 | 7 | eqeq2d 2364 | . . . . 5 0c 1c 1c 1c 0c 1c |
9 | 8 | rspcev 2955 | . . . 4 0c Nn 1c 0c 1c Nn 1c 1c |
10 | 2, 5, 9 | mp2an 653 | . . 3 Nn 1c 1c |
11 | 1, 10 | jctir 524 | . 2 Nn 1c 1c |
12 | 1cex 4142 | . . . . 5 1c | |
13 | addcexg 4393 | . . . . 5 1c 1c | |
14 | 12, 13 | mpan2 652 | . . . 4 1c |
15 | opkltfing 4449 | . . . 4 1c 1c fin Nn 1c 1c | |
16 | 14, 15 | mpdan 649 | . . 3 1c fin Nn 1c 1c |
17 | 16 | adantr 451 | . 2 1c fin Nn 1c 1c |
18 | 11, 17 | mpbird 223 | 1 1c fin |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wceq 1642 wcel 1710 wne 2516 wrex 2615 cvv 2859 c0 3550 copk 4057 1cc1c 4134 Nn cnnc 4373 0cc0c 4374 cplc 4375 fin cltfin 4433 |
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-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-pw 3724 df-sn 3741 df-pr 3742 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-p6 4191 df-sik 4192 df-ssetk 4193 df-0c 4377 df-addc 4378 df-nnc 4379 df-ltfin 4441 |
This theorem is referenced by: ltfintri 4466 |
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