Theorem tlecg 6231

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 ) → (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 6211	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c N ↔ ∃p ∈ NC N = (M +c p)))
2		tccl 6161	. . . . . . . 8 ⊢ (M ∈ NC → Tc M ∈ NC )
3		tccl 6161	. . . . . . . 8 ⊢ (p ∈ NC → Tc p ∈ NC )
4		addlecncs 6210	. . . . . . . 8 ⊢ (( Tc M ∈ NC ∧ Tc p ∈ NC ) → Tc M ≤c ( Tc M +c Tc p))
5	2, 3, 4	syl2an 463	. . . . . . 7 ⊢ ((M ∈ NC ∧ p ∈ NC ) → Tc M ≤c ( Tc M +c Tc p))
6		tcdi 6165	. . . . . . 7 ⊢ ((M ∈ NC ∧ p ∈ NC ) → Tc (M +c p) = ( Tc M +c Tc p))
7	5, 6	breqtrrd 4666	. . . . . 6 ⊢ ((M ∈ NC ∧ p ∈ NC ) → Tc M ≤c Tc (M +c p))
8		tceq 6159	. . . . . . 7 ⊢ (N = (M +c p) → Tc N = Tc (M +c p))
9	8	breq2d 4652	. . . . . 6 ⊢ (N = (M +c p) → ( Tc M ≤c Tc N ↔ Tc M ≤c Tc (M +c p)))
10	7, 9	syl5ibrcom 213	. . . . 5 ⊢ ((M ∈ NC ∧ p ∈ NC ) → (N = (M +c p) → Tc M ≤c Tc N))
11	10	rexlimdva 2739	. . . 4 ⊢ (M ∈ NC → (∃p ∈ NC N = (M +c p) → Tc M ≤c Tc N))
12	11	adantr 451	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (∃p ∈ NC N = (M +c p) → Tc M ≤c Tc N))
13	1, 12	sylbid 206	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c N → Tc M ≤c Tc N))
14		tccl 6161	. . . 4 ⊢ (N ∈ NC → Tc N ∈ NC )
15		dflec2 6211	. . . 4 ⊢ (( Tc M ∈ NC ∧ Tc N ∈ NC ) → ( Tc M ≤c Tc N ↔ ∃p ∈ NC Tc N = ( Tc M +c p)))
16	2, 14, 15	syl2an 463	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc M ≤c Tc N ↔ ∃p ∈ 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 6230	. . . . . . 7 ⊢ (((N ∈ NC ∧ M ∈ NC ∧ p ∈ NC ) ∧ Tc N = ( Tc M +c p)) → ∃q ∈ NC p = Tc q)
22	17, 18, 19, 20, 21	syl31anc 1185	. . . . . 6 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ (p ∈ NC ∧ Tc N = ( Tc M +c p))) → ∃q ∈ NC p = Tc q)
23		addceq2 4385	. . . . . . . . . . . . 13 ⊢ (p = Tc q → ( Tc M +c p) = ( Tc M +c Tc q))
24	23	eqeq2d 2364	. . . . . . . . . . . 12 ⊢ (p = Tc q → ( Tc N = ( Tc M +c p) ↔ Tc N = ( Tc M +c Tc q)))
25	24	biimpac 472	. . . . . . . . . . 11 ⊢ (( Tc N = ( Tc M +c p) ∧ p = Tc q) → Tc N = ( Tc M +c Tc q))
26		tcdi 6165	. . . . . . . . . . . . . 14 ⊢ ((M ∈ NC ∧ q ∈ NC ) → Tc (M +c q) = ( Tc M +c Tc q))
27	26	adantlr 695	. . . . . . . . . . . . 13 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → Tc (M +c q) = ( Tc M +c Tc q))
28	27	eqeq2d 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 6145	. . . . . . . . . . . . . . 15 ⊢ ((M ∈ NC ∧ q ∈ NC ) → (M +c q) ∈ NC )
31	30	adantlr 695	. . . . . . . . . . . . . 14 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → (M +c q) ∈ NC )
32		tc11 6229	. . . . . . . . . . . . . 14 ⊢ ((N ∈ NC ∧ (M +c q) ∈ NC ) → ( Tc N = Tc (M +c q) ↔ N = (M +c q)))
33	29, 31, 32	syl2anc 642	. . . . . . . . . . . . 13 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → ( Tc N = Tc (M +c q) ↔ N = (M +c q)))
34		addlecncs 6210	. . . . . . . . . . . . . . 15 ⊢ ((M ∈ NC ∧ q ∈ NC ) → M ≤c (M +c q))
35		breq2 4644	. . . . . . . . . . . . . . 15 ⊢ (N = (M +c q) → (M ≤c N ↔ M ≤c (M +c q)))
36	34, 35	syl5ibrcom 213	. . . . . . . . . . . . . 14 ⊢ ((M ∈ NC ∧ q ∈ NC ) → (N = (M +c q) → M ≤c N))
37	36	adantlr 695	. . . . . . . . . . . . 13 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → (N = (M +c q) → M ≤c N))
38	33, 37	sylbid 206	. . . . . . . . . . . 12 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → ( Tc N = Tc (M +c q) → M ≤c N))
39	28, 38	sylbird 226	. . . . . . . . . . 11 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → ( Tc N = ( Tc M +c Tc q) → M ≤c N))
40	25, 39	syl5 28	. . . . . . . . . 10 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) → (( Tc N = ( Tc M +c p) ∧ p = Tc q) → M ≤c N))
41	40	expdimp 426	. . . . . . . . 9 ⊢ ((((M ∈ NC ∧ N ∈ NC ) ∧ q ∈ NC ) ∧ Tc N = ( Tc M +c p)) → (p = Tc q → M ≤c N))
42	41	an32s 779	. . . . . . . 8 ⊢ ((((M ∈ NC ∧ N ∈ NC ) ∧ Tc N = ( Tc M +c p)) ∧ q ∈ NC ) → (p = Tc q → M ≤c N))
43	42	rexlimdva 2739	. . . . . . 7 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ Tc N = ( Tc M +c p)) → (∃q ∈ NC p = Tc q → M ≤c N))
44	43	adantrl 696	. . . . . 6 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ (p ∈ NC ∧ Tc N = ( Tc M +c p))) → (∃q ∈ NC p = Tc q → M ≤c N))
45	22, 44	mpd 14	. . . . 5 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ (p ∈ NC ∧ Tc N = ( Tc M +c p))) → M ≤c N)
46	45	expr 598	. . . 4 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ p ∈ NC ) → ( Tc N = ( Tc M +c p) → M ≤c N))
47	46	rexlimdva 2739	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (∃p ∈ NC Tc N = ( Tc M +c p) → M ≤c N))
48	16, 47	sylbid 206	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc M ≤c Tc N → M ≤c N))
49	13, 48	impbid 183	1 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c N ↔ Tc M ≤c Tc N))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   +_c cplc 4376   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090   T_c ctc 6094

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102  df-tc 6104

This theorem is referenced by:  ce2le  6234  nchoicelem9  6298  nchoicelem19  6308