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 e. NC /\ N e. NC ) -> (M <_c N <-> Tc M <_c Tc N))

Proof of Theorem tlecg

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dflec2 6210	. . 3 |- ((M e. NC /\ N e. NC ) -> (M <_c N <-> E.p e. NC N = (M +o p)))
2		tccl 6160	. . . . . . . 8 |- (M e. NC -> Tc M e. NC )
3		tccl 6160	. . . . . . . 8 |- (p e. NC -> Tc p e. NC )
4		addlecncs 6209	. . . . . . . 8 |- (( Tc M e. NC /\ Tc p e. NC ) -> Tc M <_c ( Tc M +o Tc p))
5	2, 3, 4	syl2an 463	. . . . . . 7 |- ((M e. NC /\ p e. NC ) -> Tc M <_c ( Tc M +o Tc p))
6		tcdi 6164	. . . . . . 7 |- ((M e. NC /\ p e. NC ) -> Tc (M +o p) = ( Tc M +o Tc p))
7	5, 6	breqtrrd 4665	. . . . . 6 |- ((M e. NC /\ p e. NC ) -> Tc M <_c Tc (M +o p))
8		tceq 6158	. . . . . . 7 |- (N = (M +o p) -> Tc N = Tc (M +o p))
9	8	breq2d 4651	. . . . . 6 |- (N = (M +o p) -> ( Tc M <_c Tc N <-> Tc M <_c Tc (M +o p)))
10	7, 9	syl5ibrcom 213	. . . . 5 |- ((M e. NC /\ p e. NC ) -> (N = (M +o p) -> Tc M <_c Tc N))
11	10	rexlimdva 2738	. . . 4 |- (M e. NC -> (E.p e. NC N = (M +o p) -> Tc M <_c Tc N))
12	11	adantr 451	. . 3 |- ((M e. NC /\ N e. NC ) -> (E.p e. NC N = (M +o p) -> Tc M <_c Tc N))
13	1, 12	sylbid 206	. 2 |- ((M e. NC /\ N e. NC ) -> (M <_c N -> Tc M <_c Tc N))
14		tccl 6160	. . . 4 |- (N e. NC -> Tc N e. NC )
15		dflec2 6210	. . . 4 |- (( Tc M e. NC /\ Tc N e. NC ) -> ( Tc M <_c Tc N <-> E.p e. NC Tc N = ( Tc M +o p)))
16	2, 14, 15	syl2an 463	. . 3 |- ((M e. NC /\ N e. NC ) -> ( Tc M <_c Tc N <-> E.p e. NC Tc N = ( Tc M +o p)))
17		simplr 731	. . . . . . 7 |- (((M e. NC /\ N e. NC ) /\ (p e. NC /\ Tc N = ( Tc M +o p))) -> N e. NC )
18		simpll 730	. . . . . . 7 |- (((M e. NC /\ N e. NC ) /\ (p e. NC /\ Tc N = ( Tc M +o p))) -> M e. NC )
19		simprl 732	. . . . . . 7 |- (((M e. NC /\ N e. NC ) /\ (p e. NC /\ Tc N = ( Tc M +o p))) -> p e. NC )
20		simprr 733	. . . . . . 7 |- (((M e. NC /\ N e. NC ) /\ (p e. NC /\ Tc N = ( Tc M +o p))) -> Tc N = ( Tc M +o p))
21		taddc 6229	. . . . . . 7 |- (((N e. NC /\ M e. NC /\ p e. NC ) /\ Tc N = ( Tc M +o p)) -> E.q e. NC p = Tc q)
22	17, 18, 19, 20, 21	syl31anc 1185	. . . . . 6 |- (((M e. NC /\ N e. NC ) /\ (p e. NC /\ Tc N = ( Tc M +o p))) -> E.q e. NC p = Tc q)
23		addceq2 4384	. . . . . . . . . . . . 13 |- (p = Tc q -> ( Tc M +o p) = ( Tc M +o Tc q))
24	23	eqeq2d 2364	. . . . . . . . . . . 12 |- (p = Tc q -> ( Tc N = ( Tc M +o p) <-> Tc N = ( Tc M +o Tc q)))
25	24	biimpac 472	. . . . . . . . . . 11 |- (( Tc N = ( Tc M +o p) /\ p = Tc q) -> Tc N = ( Tc M +o Tc q))
26		tcdi 6164	. . . . . . . . . . . . . 14 |- ((M e. NC /\ q e. NC ) -> Tc (M +o q) = ( Tc M +o Tc q))
27	26	adantlr 695	. . . . . . . . . . . . 13 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> Tc (M +o q) = ( Tc M +o Tc q))
28	27	eqeq2d 2364	. . . . . . . . . . . 12 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> ( Tc N = Tc (M +o q) <-> Tc N = ( Tc M +o Tc q)))
29		simplr 731	. . . . . . . . . . . . . 14 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> N e. NC )
30		ncaddccl 6144	. . . . . . . . . . . . . . 15 |- ((M e. NC /\ q e. NC ) -> (M +o q) e. NC )
31	30	adantlr 695	. . . . . . . . . . . . . 14 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> (M +o q) e. NC )
32		tc11 6228	. . . . . . . . . . . . . 14 |- ((N e. NC /\ (M +o q) e. NC ) -> ( Tc N = Tc (M +o q) <-> N = (M +o q)))
33	29, 31, 32	syl2anc 642	. . . . . . . . . . . . 13 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> ( Tc N = Tc (M +o q) <-> N = (M +o q)))
34		addlecncs 6209	. . . . . . . . . . . . . . 15 |- ((M e. NC /\ q e. NC ) -> M <_c (M +o q))
35		breq2 4643	. . . . . . . . . . . . . . 15 |- (N = (M +o q) -> (M <_c N <-> M <_c (M +o q)))
36	34, 35	syl5ibrcom 213	. . . . . . . . . . . . . 14 |- ((M e. NC /\ q e. NC ) -> (N = (M +o q) -> M <_c N))
37	36	adantlr 695	. . . . . . . . . . . . 13 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> (N = (M +o q) -> M <_c N))
38	33, 37	sylbid 206	. . . . . . . . . . . 12 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> ( Tc N = Tc (M +o q) -> M <_c N))
39	28, 38	sylbird 226	. . . . . . . . . . 11 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> ( Tc N = ( Tc M +o Tc q) -> M <_c N))
40	25, 39	syl5 28	. . . . . . . . . 10 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> (( Tc N = ( Tc M +o p) /\ p = Tc q) -> M <_c N))
41	40	expdimp 426	. . . . . . . . 9 |- ((((M e. NC /\ N e. NC ) /\ q e. NC ) /\ Tc N = ( Tc M +o p)) -> (p = Tc q -> M <_c N))
42	41	an32s 779	. . . . . . . 8 |- ((((M e. NC /\ N e. NC ) /\ Tc N = ( Tc M +o p)) /\ q e. NC ) -> (p = Tc q -> M <_c N))
43	42	rexlimdva 2738	. . . . . . 7 |- (((M e. NC /\ N e. NC ) /\ Tc N = ( Tc M +o p)) -> (E.q e. NC p = Tc q -> M <_c N))
44	43	adantrl 696	. . . . . 6 |- (((M e. NC /\ N e. NC ) /\ (p e. NC /\ Tc N = ( Tc M +o p))) -> (E.q e. NC p = Tc q -> M <_c N))
45	22, 44	mpd 14	. . . . 5 |- (((M e. NC /\ N e. NC ) /\ (p e. NC /\ Tc N = ( Tc M +o p))) -> M <_c N)
46	45	expr 598	. . . 4 |- (((M e. NC /\ N e. NC ) /\ p e. NC ) -> ( Tc N = ( Tc M +o p) -> M <_c N))
47	46	rexlimdva 2738	. . 3 |- ((M e. NC /\ N e. NC ) -> (E.p e. NC Tc N = ( Tc M +o p) -> M <_c N))
48	16, 47	sylbid 206	. 2 |- ((M e. NC /\ N e. NC ) -> ( Tc M <_c Tc N -> M <_c N))
49	13, 48	impbid 183	1 |- ((M e. NC /\ N e. NC ) -> (M <_c N <-> Tc M <_c Tc N))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615   +o cplc 4375   class class class wbr 4639   NC cncs 6088   <__c clec 6089   T_c 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-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