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

Theorem tlecg 6230
 Description: T-raising perserves order for cardinals. Theorem 5.5 of [Specker] p. 973. (Contributed by SF, 11-Mar-2015.)
Assertion
Ref Expression
tlecg ((M NC N NC ) → (Mc NTc Mc Tc N))

Proof of Theorem tlecg
Dummy variables p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dflec2 6210 . . 3 ((M NC N NC ) → (Mc Np NC N = (M +c p)))
2 tccl 6160 . . . . . . . 8 (M NCTc M NC )
3 tccl 6160 . . . . . . . 8 (p NCTc p NC )
4 addlecncs 6209 . . . . . . . 8 (( Tc M NC Tc p NC ) → Tc Mc ( Tc M +c Tc p))
52, 3, 4syl2an 463 . . . . . . 7 ((M NC p NC ) → Tc Mc ( Tc M +c Tc p))
6 tcdi 6164 . . . . . . 7 ((M NC p NC ) → Tc (M +c p) = ( Tc M +c Tc p))
75, 6breqtrrd 4665 . . . . . 6 ((M NC p NC ) → Tc Mc Tc (M +c p))
8 tceq 6158 . . . . . . 7 (N = (M +c p) → Tc N = Tc (M +c p))
98breq2d 4651 . . . . . 6 (N = (M +c p) → ( Tc Mc Tc NTc Mc Tc (M +c p)))
107, 9syl5ibrcom 213 . . . . 5 ((M NC p NC ) → (N = (M +c p) → Tc Mc Tc N))
1110rexlimdva 2738 . . . 4 (M NC → (p NC N = (M +c p) → Tc Mc Tc N))
1211adantr 451 . . 3 ((M NC N NC ) → (p NC N = (M +c p) → Tc Mc Tc N))
131, 12sylbid 206 . 2 ((M NC N NC ) → (Mc NTc Mc Tc N))
14 tccl 6160 . . . 4 (N NCTc N NC )
15 dflec2 6210 . . . 4 (( Tc M NC Tc N NC ) → ( Tc Mc Tc Np NC Tc N = ( Tc M +c p)))
162, 14, 15syl2an 463 . . 3 ((M NC N NC ) → ( Tc Mc Tc Np NC Tc N = ( Tc M +c p)))
17 simplr 731 . . . . . . 7 (((M NC N NC ) (p NC Tc N = ( Tc M +c p))) → N NC )
18 simpll 730 . . . . . . 7 (((M NC N NC ) (p NC Tc N = ( Tc M +c p))) → M NC )
19 simprl 732 . . . . . . 7 (((M NC N NC ) (p NC Tc N = ( Tc M +c p))) → p NC )
20 simprr 733 . . . . . . 7 (((M NC N NC ) (p NC Tc N = ( Tc M +c p))) → Tc N = ( Tc M +c p))
21 taddc 6229 . . . . . . 7 (((N NC M NC p NC ) Tc N = ( Tc M +c p)) → q NC p = Tc q)
2217, 18, 19, 20, 21syl31anc 1185 . . . . . 6 (((M NC N NC ) (p NC Tc N = ( Tc M +c p))) → q NC p = Tc q)
23 addceq2 4384 . . . . . . . . . . . . 13 (p = Tc q → ( Tc M +c p) = ( Tc M +c Tc q))
2423eqeq2d 2364 . . . . . . . . . . . 12 (p = Tc q → ( Tc N = ( Tc M +c p) ↔ Tc N = ( Tc M +c Tc q)))
2524biimpac 472 . . . . . . . . . . 11 (( Tc N = ( Tc M +c p) p = Tc q) → Tc N = ( Tc M +c Tc q))
26 tcdi 6164 . . . . . . . . . . . . . 14 ((M NC q NC ) → Tc (M +c q) = ( Tc M +c Tc q))
2726adantlr 695 . . . . . . . . . . . . 13 (((M NC N NC ) q NC ) → Tc (M +c q) = ( Tc M +c Tc q))
2827eqeq2d 2364 . . . . . . . . . . . 12 (((M NC N NC ) q NC ) → ( Tc N = Tc (M +c q) ↔ Tc N = ( Tc M +c Tc q)))
29 simplr 731 . . . . . . . . . . . . . 14 (((M NC N NC ) q NC ) → N NC )
30 ncaddccl 6144 . . . . . . . . . . . . . . 15 ((M NC q NC ) → (M +c q) NC )
3130adantlr 695 . . . . . . . . . . . . . 14 (((M NC N NC ) q NC ) → (M +c q) NC )
32 tc11 6228 . . . . . . . . . . . . . 14 ((N NC (M +c q) NC ) → ( Tc N = Tc (M +c q) ↔ N = (M +c q)))
3329, 31, 32syl2anc 642 . . . . . . . . . . . . 13 (((M NC N NC ) q NC ) → ( Tc N = Tc (M +c q) ↔ N = (M +c q)))
34 addlecncs 6209 . . . . . . . . . . . . . . 15 ((M NC q NC ) → Mc (M +c q))
35 breq2 4643 . . . . . . . . . . . . . . 15 (N = (M +c q) → (Mc NMc (M +c q)))
3634, 35syl5ibrcom 213 . . . . . . . . . . . . . 14 ((M NC q NC ) → (N = (M +c q) → Mc N))
3736adantlr 695 . . . . . . . . . . . . 13 (((M NC N NC ) q NC ) → (N = (M +c q) → Mc N))
3833, 37sylbid 206 . . . . . . . . . . . 12 (((M NC N NC ) q NC ) → ( Tc N = Tc (M +c q) → Mc N))
3928, 38sylbird 226 . . . . . . . . . . 11 (((M NC N NC ) q NC ) → ( Tc N = ( Tc M +c Tc q) → Mc N))
4025, 39syl5 28 . . . . . . . . . 10 (((M NC N NC ) q NC ) → (( Tc N = ( Tc M +c p) p = Tc q) → Mc N))
4140expdimp 426 . . . . . . . . 9 ((((M NC N NC ) q NC ) Tc N = ( Tc M +c p)) → (p = Tc qMc N))
4241an32s 779 . . . . . . . 8 ((((M NC N NC ) Tc N = ( Tc M +c p)) q NC ) → (p = Tc qMc N))
4342rexlimdva 2738 . . . . . . 7 (((M NC N NC ) Tc N = ( Tc M +c p)) → (q NC p = Tc qMc N))
4443adantrl 696 . . . . . 6 (((M NC N NC ) (p NC Tc N = ( Tc M +c p))) → (q NC p = Tc qMc N))
4522, 44mpd 14 . . . . 5 (((M NC N NC ) (p NC Tc N = ( Tc M +c p))) → Mc N)
4645expr 598 . . . 4 (((M NC N NC ) p NC ) → ( Tc N = ( Tc M +c p) → Mc N))
4746rexlimdva 2738 . . 3 ((M NC N NC ) → (p NC Tc N = ( Tc M +c p) → Mc N))
4816, 47sylbid 206 . 2 ((M NC N NC ) → ( Tc Mc Tc NMc N))
4913, 48impbid 183 1 ((M NC N NC ) → (Mc NTc Mc Tc N))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   +c cplc 4375   class class class wbr 4639   NC cncs 6088   ≤c clec 6089   Tc ctc 6093 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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  df-tc 6103 This theorem is referenced by:  ce2le  6233  nchoicelem9  6297  nchoicelem19  6307
 Copyright terms: Public domain W3C validator