New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  nmembers1 Unicode version

Theorem nmembers1 6271
 Description: Count the number of elements in a natural interval. From nmembers1lem2 6269 and nmembers1lem3 6270, we would expect to arrive at Nn 1c c c , but this proposition is not stratifiable. Instead, we arrive at the weaker conclusion below. We can arrive at the earlier proposition once we add the Axiom of Counting, which we will do later. (Contributed by Scott Fenton, 3-Aug-2019.)
Assertion
Ref Expression
nmembers1 Nn Nn 1c c c Tc Tc
Distinct variable group:   ,

Proof of Theorem nmembers1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmembers1lem1 6268 . 2 Nn 1c c c Tc Tc
2 breq2 4643 . . . . 5 0c c c 0c
32anbi2d 684 . . . 4 0c 1c c c 1c c c 0c
43rabbidv 2851 . . 3 0c Nn 1c c c Nn 1c c c 0c
5 tceq 6158 . . . 4 0c Tc Tc 0c
6 tceq 6158 . . . 4 Tc Tc 0c Tc Tc Tc Tc 0c
75, 6syl 15 . . 3 0c Tc Tc Tc Tc 0c
84, 7eleq12d 2421 . 2 0c Nn 1c c c Tc Tc Nn 1c c c 0c Tc Tc 0c
9 breq2 4643 . . . . 5 c c
109anbi2d 684 . . . 4 1c c c 1c c c
1110rabbidv 2851 . . 3 Nn 1c c c Nn 1c c c
12 tceq 6158 . . . 4 Tc Tc
13 tceq 6158 . . . 4 Tc Tc Tc Tc Tc Tc
1412, 13syl 15 . . 3 Tc Tc Tc Tc
1511, 14eleq12d 2421 . 2 Nn 1c c c Tc Tc Nn 1c c c Tc Tc
16 breq2 4643 . . . . 5 1c c c 1c
1716anbi2d 684 . . . 4 1c 1c c c 1c c c 1c
1817rabbidv 2851 . . 3 1c Nn 1c c c Nn 1c c c 1c
19 tceq 6158 . . . 4 1c Tc Tc 1c
20 tceq 6158 . . . 4 Tc Tc 1c Tc Tc Tc Tc 1c
2119, 20syl 15 . . 3 1c Tc Tc Tc Tc 1c
2218, 21eleq12d 2421 . 2 1c Nn 1c c c Tc Tc Nn 1c c c 1c Tc Tc 1c
23 breq2 4643 . . . . 5 c c
2423anbi2d 684 . . . 4 1c c c 1c c c
2524rabbidv 2851 . . 3 Nn 1c c c Nn 1c c c
26 tceq 6158 . . . 4 Tc Tc
27 tceq 6158 . . . 4 Tc Tc Tc Tc Tc Tc
2826, 27syl 15 . . 3 Tc Tc Tc Tc
2925, 28eleq12d 2421 . 2 Nn 1c c c Tc Tc Nn 1c c c Tc Tc
30 nmembers1lem2 6269 . . 3 Nn 1c c c 0c 0c
31 tc0c 6163 . . . . 5 Tc 0c 0c
32 tceq 6158 . . . . 5 Tc 0c 0c Tc Tc 0c Tc 0c
3331, 32ax-mp 5 . . . 4 Tc Tc 0c Tc 0c
3433, 31eqtri 2373 . . 3 Tc Tc 0c 0c
3530, 34eleqtrri 2426 . 2 Nn 1c c c 0c Tc Tc 0c
36 nntccl 6170 . . . 4 Nn Tc Nn
37 nntccl 6170 . . . 4 Tc Nn Tc Tc Nn
3836, 37syl 15 . . 3 Nn Tc Tc Nn
39 nmembers1lem3 6270 . . . 4 Nn Tc Tc Nn Nn 1c c c Tc Tc Nn 1c c c 1c Tc Tc 1c
40 nnnc 6146 . . . . . . 7 Nn NC
41 1cnc 6139 . . . . . . . . . . 11 1c NC
42 tcdi 6164 . . . . . . . . . . 11 NC 1c NC Tc 1c Tc Tc 1c
4341, 42mpan2 652 . . . . . . . . . 10 NC Tc 1c Tc Tc 1c
44 tc1c 6165 . . . . . . . . . . 11 Tc 1c 1c
4544addceq2i 4387 . . . . . . . . . 10 Tc Tc 1c Tc 1c
4643, 45syl6eq 2401 . . . . . . . . 9 NC Tc 1c Tc 1c
47 tceq 6158 . . . . . . . . 9 Tc 1c Tc 1c Tc Tc 1c Tc Tc 1c
4846, 47syl 15 . . . . . . . 8 NC Tc Tc 1c Tc Tc 1c
49 tccl 6160 . . . . . . . . . 10 NC Tc NC
50 tcdi 6164 . . . . . . . . . . 11 Tc NC 1c NC Tc Tc 1c Tc Tc Tc 1c
5141, 50mpan2 652 . . . . . . . . . 10 Tc NC Tc Tc 1c Tc Tc Tc 1c
5249, 51syl 15 . . . . . . . . 9 NC Tc Tc 1c Tc Tc Tc 1c
5344addceq2i 4387 . . . . . . . . 9 Tc Tc Tc 1c Tc Tc 1c
5452, 53syl6eq 2401 . . . . . . . 8 NC Tc Tc 1c Tc Tc 1c
5548, 54eqtrd 2385 . . . . . . 7 NC Tc Tc 1c Tc Tc 1c
5640, 55syl 15 . . . . . 6 Nn Tc Tc 1c Tc Tc 1c
5756eleq2d 2420 . . . . 5 Nn Nn 1c c c 1c Tc Tc 1c Nn 1c c c 1c Tc Tc 1c
5857adantr 451 . . . 4 Nn Tc Tc Nn Nn 1c c c 1c Tc Tc 1c Nn 1c c c 1c Tc Tc 1c
5939, 58sylibrd 225 . . 3 Nn Tc Tc Nn Nn 1c c c Tc Tc Nn 1c c c 1c Tc Tc 1c
6038, 59mpdan 649 . 2 Nn Nn 1c c c Tc Tc Nn 1c c c 1c Tc Tc 1c
611, 8, 15, 22, 29, 35, 60finds 4411 1 Nn Nn 1c c c Tc Tc
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  crab 2618  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   cplc 4375   class class class wbr 4639   NC cncs 6088   c clec 6089   Tc ctc 6093 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-pw1fn 5766  df-clos1 5873  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  df-tc 6103  df-tcfn 6107 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator