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Mirrors > Home > NFE Home > Th. List > nnltp1c | Unicode version |
Description: Any natural is less than one plus itself. (Contributed by SF, 25-Mar-2015.) |
Ref | Expression |
---|---|
nnltp1c | Nn c 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnltp1clem1 6261 | . 2 c 1c | |
2 | id 19 | . . 3 0c 0c | |
3 | addceq1 4383 | . . 3 0c 1c 0c 1c | |
4 | 2, 3 | breq12d 4652 | . 2 0c c 1c 0c c 0c 1c |
5 | id 19 | . . 3 | |
6 | addceq1 4383 | . . 3 1c 1c | |
7 | 5, 6 | breq12d 4652 | . 2 c 1c c 1c |
8 | id 19 | . . 3 1c 1c | |
9 | addceq1 4383 | . . 3 1c 1c 1c 1c | |
10 | 8, 9 | breq12d 4652 | . 2 1c c 1c 1c c 1c 1c |
11 | id 19 | . . 3 | |
12 | addceq1 4383 | . . 3 1c 1c | |
13 | 11, 12 | breq12d 4652 | . 2 c 1c c 1c |
14 | 0cnc 6138 | . . . 4 0c NC | |
15 | 1cnc 6139 | . . . 4 1c NC | |
16 | addlecncs 6209 | . . . 4 0c NC 1c NC 0c c 0c 1c | |
17 | 14, 15, 16 | mp2an 653 | . . 3 0c c 0c 1c |
18 | 0cnsuc 4401 | . . . 4 0c 1c 0c | |
19 | 18 | necomi 2598 | . . 3 0c 0c 1c |
20 | brltc 6114 | . . 3 0c c 0c 1c 0c c 0c 1c 0c 0c 1c | |
21 | 17, 19, 20 | mpbir2an 886 | . 2 0c c 0c 1c |
22 | nnnc 6146 | . . . . 5 Nn NC | |
23 | peano2nc 6145 | . . . . . 6 NC 1c NC | |
24 | 22, 23 | syl 15 | . . . . 5 Nn 1c NC |
25 | leaddc1 6214 | . . . . . . 7 NC 1c NC 1c NC c 1c 1c c 1c 1c | |
26 | 25 | ex 423 | . . . . . 6 NC 1c NC 1c NC c 1c 1c c 1c 1c |
27 | 15, 26 | mp3an3 1266 | . . . . 5 NC 1c NC c 1c 1c c 1c 1c |
28 | 22, 24, 27 | syl2anc 642 | . . . 4 Nn c 1c 1c c 1c 1c |
29 | peano2 4403 | . . . . . . 7 Nn 1c Nn | |
30 | suc11nnc 4558 | . . . . . . 7 Nn 1c Nn 1c 1c 1c 1c | |
31 | 29, 30 | mpdan 649 | . . . . . 6 Nn 1c 1c 1c 1c |
32 | 31 | biimpd 198 | . . . . 5 Nn 1c 1c 1c 1c |
33 | 32 | necon3d 2554 | . . . 4 Nn 1c 1c 1c 1c |
34 | 28, 33 | anim12d 546 | . . 3 Nn c 1c 1c 1c c 1c 1c 1c 1c 1c |
35 | brltc 6114 | . . 3 c 1c c 1c 1c | |
36 | brltc 6114 | . . 3 1c c 1c 1c 1c c 1c 1c 1c 1c 1c | |
37 | 34, 35, 36 | 3imtr4g 261 | . 2 Nn c 1c 1c c 1c 1c |
38 | 1, 4, 7, 10, 13, 21, 37 | finds 4411 | 1 Nn c 1c |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 w3a 934 wceq 1642 wcel 1710 wne 2516 1cc1c 4134 Nn cnnc 4373 0cc0c 4374 cplc 4375 class class class wbr 4639 NC cncs 6088 c clec 6089 c cltc 6090 |
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-csb 3137 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-iun 3971 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-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-fix 5740 df-cup 5742 df-disj 5744 df-addcfn 5746 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: nmembers1lem3 6270 |
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