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Mirrors > Home > NFE Home > Th. List > ncslesuc | Unicode version |
Description: Relationship between successor and cardinal less than or equal. (Contributed by Scott Fenton, 3-Aug-2019.) |
Ref | Expression |
---|---|
ncslesuc | NC NC c 1c c 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2nc 6145 | . . . 4 NC 1c NC | |
2 | dflec2 6210 | . . . 4 NC 1c NC c 1c NC 1c | |
3 | 1, 2 | sylan2 460 | . . 3 NC NC c 1c NC 1c |
4 | nc0suc 6217 | . . . . 5 NC 0c NC 1c | |
5 | addceq2 4384 | . . . . . . . . . 10 0c 0c | |
6 | addcid1 4405 | . . . . . . . . . 10 0c | |
7 | 5, 6 | syl6eq 2401 | . . . . . . . . 9 0c |
8 | 7 | eqeq2d 2364 | . . . . . . . 8 0c 1c 1c |
9 | olc 373 | . . . . . . . . 9 1c c 1c | |
10 | 9 | eqcoms 2356 | . . . . . . . 8 1c c 1c |
11 | 8, 10 | syl6bi 219 | . . . . . . 7 0c 1c c 1c |
12 | 11 | a1i 10 | . . . . . 6 NC NC 0c 1c c 1c |
13 | addceq2 4384 | . . . . . . . . . . . 12 1c 1c | |
14 | addcass 4415 | . . . . . . . . . . . 12 1c 1c | |
15 | 13, 14 | syl6eqr 2403 | . . . . . . . . . . 11 1c 1c |
16 | 15 | eqeq2d 2364 | . . . . . . . . . 10 1c 1c 1c 1c |
17 | 16 | biimpa 470 | . . . . . . . . 9 1c 1c 1c 1c |
18 | simplr 731 | . . . . . . . . . . 11 NC NC NC NC | |
19 | ncaddccl 6144 | . . . . . . . . . . . 12 NC NC NC | |
20 | 19 | adantlr 695 | . . . . . . . . . . 11 NC NC NC NC |
21 | peano4nc 6150 | . . . . . . . . . . 11 NC NC 1c 1c | |
22 | 18, 20, 21 | syl2anc 642 | . . . . . . . . . 10 NC NC NC 1c 1c |
23 | addlecncs 6209 | . . . . . . . . . . . . 13 NC NC c | |
24 | 23 | adantlr 695 | . . . . . . . . . . . 12 NC NC NC c |
25 | breq2 4643 | . . . . . . . . . . . 12 c c | |
26 | 24, 25 | syl5ibrcom 213 | . . . . . . . . . . 11 NC NC NC c |
27 | orc 374 | . . . . . . . . . . 11 c c 1c | |
28 | 26, 27 | syl6 29 | . . . . . . . . . 10 NC NC NC c 1c |
29 | 22, 28 | sylbid 206 | . . . . . . . . 9 NC NC NC 1c 1c c 1c |
30 | 17, 29 | syl5 28 | . . . . . . . 8 NC NC NC 1c 1c c 1c |
31 | 30 | exp3a 425 | . . . . . . 7 NC NC NC 1c 1c c 1c |
32 | 31 | rexlimdva 2738 | . . . . . 6 NC NC NC 1c 1c c 1c |
33 | 12, 32 | jaod 369 | . . . . 5 NC NC 0c NC 1c 1c c 1c |
34 | 4, 33 | syl5 28 | . . . 4 NC NC NC 1c c 1c |
35 | 34 | rexlimdv 2737 | . . 3 NC NC NC 1c c 1c |
36 | 3, 35 | sylbid 206 | . 2 NC NC c 1c c 1c |
37 | 1cnc 6139 | . . . . . 6 1c NC | |
38 | addlecncs 6209 | . . . . . 6 NC 1c NC c 1c | |
39 | 37, 38 | mpan2 652 | . . . . 5 NC c 1c |
40 | 39 | adantl 452 | . . . 4 NC NC c 1c |
41 | 1 | adantl 452 | . . . . 5 NC NC 1c NC |
42 | lectr 6211 | . . . . 5 NC NC 1c NC c c 1c c 1c | |
43 | 41, 42 | mpd3an3 1278 | . . . 4 NC NC c c 1c c 1c |
44 | 40, 43 | mpan2d 655 | . . 3 NC NC c c 1c |
45 | nclecid 6197 | . . . . . 6 1c NC 1c c 1c | |
46 | 1, 45 | syl 15 | . . . . 5 NC 1c c 1c |
47 | 46 | adantl 452 | . . . 4 NC NC 1c c 1c |
48 | breq1 4642 | . . . 4 1c c 1c 1c c 1c | |
49 | 47, 48 | syl5ibrcom 213 | . . 3 NC NC 1c c 1c |
50 | 44, 49 | jaod 369 | . 2 NC NC c 1c c 1c |
51 | 36, 50 | impbid 183 | 1 NC NC c 1c c 1c |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wo 357 wa 358 wceq 1642 wcel 1710 wrex 2615 1cc1c 4134 0cc0c 4374 cplc 4375 class class class wbr 4639 NC cncs 6088 c clec 6089 |
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-nc 6101 |
This theorem is referenced by: nmembers1lem3 6270 |
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