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Theorem ncslesuc 6267
 Description: Relationship between successor and cardinal less than or equal. (Contributed by Scott Fenton, 3-Aug-2019.)
Assertion
Ref Expression
ncslesuc NC NC c 1c c 1c

Proof of Theorem ncslesuc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano2nc 6145 . . . 4 NC 1c NC
2 dflec2 6210 . . . 4 NC 1c NC c 1c NC 1c
31, 2sylan2 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
75, 6syl6eq 2401 . . . . . . . . 9 0c
87eqeq2d 2364 . . . . . . . 8 0c 1c 1c
9 olc 373 . . . . . . . . 9 1c c 1c
109eqcoms 2356 . . . . . . . 8 1c c 1c
118, 10syl6bi 219 . . . . . . 7 0c 1c c 1c
1211a1i 10 . . . . . 6 NC NC 0c 1c c 1c
13 addceq2 4384 . . . . . . . . . . . 12 1c 1c
14 addcass 4415 . . . . . . . . . . . 12 1c 1c
1513, 14syl6eqr 2403 . . . . . . . . . . 11 1c 1c
1615eqeq2d 2364 . . . . . . . . . 10 1c 1c 1c 1c
1716biimpa 470 . . . . . . . . 9 1c 1c 1c 1c
18 simplr 731 . . . . . . . . . . 11 NC NC NC NC
19 ncaddccl 6144 . . . . . . . . . . . 12 NC NC NC
2019adantlr 695 . . . . . . . . . . 11 NC NC NC NC
21 peano4nc 6150 . . . . . . . . . . 11 NC NC 1c 1c
2218, 20, 21syl2anc 642 . . . . . . . . . 10 NC NC NC 1c 1c
23 addlecncs 6209 . . . . . . . . . . . . 13 NC NC c
2423adantlr 695 . . . . . . . . . . . 12 NC NC NC c
25 breq2 4643 . . . . . . . . . . . 12 c c
2624, 25syl5ibrcom 213 . . . . . . . . . . 11 NC NC NC c
27 orc 374 . . . . . . . . . . 11 c c 1c
2826, 27syl6 29 . . . . . . . . . 10 NC NC NC c 1c
2922, 28sylbid 206 . . . . . . . . 9 NC NC NC 1c 1c c 1c
3017, 29syl5 28 . . . . . . . 8 NC NC NC 1c 1c c 1c
3130exp3a 425 . . . . . . 7 NC NC NC 1c 1c c 1c
3231rexlimdva 2738 . . . . . 6 NC NC NC 1c 1c c 1c
3312, 32jaod 369 . . . . 5 NC NC 0c NC 1c 1c c 1c
344, 33syl5 28 . . . 4 NC NC NC 1c c 1c
3534rexlimdv 2737 . . 3 NC NC NC 1c c 1c
363, 35sylbid 206 . 2 NC NC c 1c c 1c
37 1cnc 6139 . . . . . 6 1c NC
38 addlecncs 6209 . . . . . 6 NC 1c NC c 1c
3937, 38mpan2 652 . . . . 5 NC c 1c
4039adantl 452 . . . 4 NC NC c 1c
411adantl 452 . . . . 5 NC NC 1c NC
42 lectr 6211 . . . . 5 NC NC 1c NC c c 1c c 1c
4341, 42mpd3an3 1278 . . . 4 NC NC c c 1c c 1c
4440, 43mpan2d 655 . . 3 NC NC c c 1c
45 nclecid 6197 . . . . . 6 1c NC 1c c 1c
461, 45syl 15 . . . . 5 NC 1c c 1c
4746adantl 452 . . . 4 NC NC 1c c 1c
48 breq1 4642 . . . 4 1c c 1c 1c c 1c
4947, 48syl5ibrcom 213 . . 3 NC NC 1c c 1c
5044, 49jaod 369 . 2 NC NC c 1c c 1c
5136, 50impbid 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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