Theorem letc 6232

Description: If a cardinal is less than or equal to a T-raising, then it is also a T-raising. Theorem 5.6 of [Specker] p. 973. (Contributed by SF, 11-Mar-2015.)

Assertion

Ref	Expression
letc	|- ((M e. NC /\ N e. NC /\ M <_c Tc N) -> E.p e. NC M = Tc p)

Distinct variable group:   M,p

Allowed substitution hint:   N(p)

Proof of Theorem letc

Dummy variables a b c m n q x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tccl 6161	. . . 4 |- (N e. NC -> Tc N e. NC )
2		dflec2 6211	. . . 4 |- ((M e. NC /\ Tc N e. NC ) -> (M <_c Tc N <-> E.q e. NC Tc N = (M +o q)))
3	1, 2	sylan2 460	. . 3 |- ((M e. NC /\ N e. NC ) -> (M <_c Tc N <-> E.q e. NC Tc N = (M +o q)))
4		elncs 6120	. . . . . . . 8 |- (M e. NC <-> E.a M = Nc a)
5		elncs 6120	. . . . . . . 8 |- (N e. NC <-> E.b N = Nc b)
6		elncs 6120	. . . . . . . 8 |- (q e. NC <-> E.c q = Nc c)
7	4, 5, 6	3anbi123i 1140	. . . . . . 7 |- ((M e. NC /\ N e. NC /\ q e. NC ) <-> (E.a M = Nc a /\ E.b N = Nc b /\ E.c q = Nc c))
8		eeeanv 1914	. . . . . . 7 |- (E.aE.bE.c(M = Nc a /\ N = Nc b /\ q = Nc c) <-> (E.a M = Nc a /\ E.b N = Nc b /\ E.c q = Nc c))
9	7, 8	bitr4i 243	. . . . . 6 |- ((M e. NC /\ N e. NC /\ q e. NC ) <-> E.aE.bE.c(M = Nc a /\ N = Nc b /\ q = Nc c))
10		eqcom 2355	. . . . . . . . . . 11 |- ( Nc ~P1 b = ( Nc a +o Nc c) <-> ( Nc a +o Nc c) = Nc ~P1 b)
11		vex 2863	. . . . . . . . . . . . . 14 |- a e. _V
12	11	ncelncsi 6122	. . . . . . . . . . . . 13 |- Nc a e. NC
13		vex 2863	. . . . . . . . . . . . . 14 |- c e. _V
14	13	ncelncsi 6122	. . . . . . . . . . . . 13 |- Nc c e. NC
15		ncaddccl 6145	. . . . . . . . . . . . 13 |- (( Nc a e. NC /\ Nc c e. NC ) -> ( Nc a +o Nc c) e. NC )
16	12, 14, 15	mp2an 653	. . . . . . . . . . . 12 |- ( Nc a +o Nc c) e. NC
17		ncseqnc 6129	. . . . . . . . . . . 12 |- (( Nc a +o Nc c) e. NC -> (( Nc a +o Nc c) = Nc ~P1 b <-> ~P1 b e. ( Nc a +o Nc c)))
18	16, 17	ax-mp 5	. . . . . . . . . . 11 |- (( Nc a +o Nc c) = Nc ~P1 b <-> ~P1 b e. ( Nc a +o Nc c))
19	10, 18	bitri 240	. . . . . . . . . 10 |- ( Nc ~P1 b = ( Nc a +o Nc c) <-> ~P1 b e. ( Nc a +o Nc c))
20		eladdc 4399	. . . . . . . . . . 11 |- (~P1 b e. ( Nc a +o Nc c) <-> E.x e. Nc aE.y e. Nc c((x i^i y) = (/) /\ ~P1 b = (x u. y)))
21		vex 2863	. . . . . . . . . . . . . . 15 |- x e. _V
22		vex 2863	. . . . . . . . . . . . . . 15 |- y e. _V
23	21, 22	pw1equn 4332	. . . . . . . . . . . . . 14 |- (~P1 b = (x u. y) <-> E.nE.m(b = (n u. m) /\ x = ~P1 n /\ y = ~P1 m))
24		eleq1 2413	. . . . . . . . . . . . . . . . . . . 20 |- (x = ~P1 n -> (x e. Nc a <-> ~P1 n e. Nc a))
25		eleq1 2413	. . . . . . . . . . . . . . . . . . . 20 |- (y = ~P1 m -> (y e. Nc c <-> ~P1 m e. Nc c))
26	24, 25	bi2anan9 843	. . . . . . . . . . . . . . . . . . 19 |- ((x = ~P1 n /\ y = ~P1 m) -> ((x e. Nc a /\ y e. Nc c) <-> (~P1 n e. Nc a /\ ~P1 m e. Nc c)))
27		ineq12 3453	. . . . . . . . . . . . . . . . . . . 20 |- ((x = ~P1 n /\ y = ~P1 m) -> (x i^i y) = (~P1 n i^i ~P1 m))
28	27	eqeq1d 2361	. . . . . . . . . . . . . . . . . . 19 |- ((x = ~P1 n /\ y = ~P1 m) -> ((x i^i y) = (/) <-> (~P1 n i^i ~P1 m) = (/)))
29	26, 28	anbi12d 691	. . . . . . . . . . . . . . . . . 18 |- ((x = ~P1 n /\ y = ~P1 m) -> (((x e. Nc a /\ y e. Nc c) /\ (x i^i y) = (/)) <-> ((~P1 n e. Nc a /\ ~P1 m e. Nc c) /\ (~P1 n i^i ~P1 m) = (/))))
30		ncseqnc 6129	. . . . . . . . . . . . . . . . . . . . 21 |- ( Nc a e. NC -> ( Nc a = Nc ~P1 n <-> ~P1 n e. Nc a))
31	12, 30	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( Nc a = Nc ~P1 n <-> ~P1 n e. Nc a)
32		vex 2863	. . . . . . . . . . . . . . . . . . . . . 22 |- n e. _V
33	32	ncelncsi 6122	. . . . . . . . . . . . . . . . . . . . 21 |- Nc n e. NC
34		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (p = Nc n -> Tc p = Tc Nc n)
35	32	tcnc 6226	. . . . . . . . . . . . . . . . . . . . . . . 24 |- Tc Nc n = Nc ~P1 n
36	34, 35	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . . . 23 |- (p = Nc n -> Tc p = Nc ~P1 n)
37	36	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . . 22 |- (p = Nc n -> ( Nc a = Tc p <-> Nc a = Nc ~P1 n))
38	37	rspcev 2956	. . . . . . . . . . . . . . . . . . . . 21 |- (( Nc n e. NC /\ Nc a = Nc ~P1 n) -> E.p e. NC Nc a = Tc p)
39	33, 38	mpan 651	. . . . . . . . . . . . . . . . . . . 20 |- ( Nc a = Nc ~P1 n -> E.p e. NC Nc a = Tc p)
40	31, 39	sylbir 204	. . . . . . . . . . . . . . . . . . 19 |- (~P1 n e. Nc a -> E.p e. NC Nc a = Tc p)
41	40	ad2antrr 706	. . . . . . . . . . . . . . . . . 18 |- (((~P1 n e. Nc a /\ ~P1 m e. Nc c) /\ (~P1 n i^i ~P1 m) = (/)) -> E.p e. NC Nc a = Tc p)
42	29, 41	syl6bi 219	. . . . . . . . . . . . . . . . 17 |- ((x = ~P1 n /\ y = ~P1 m) -> (((x e. Nc a /\ y e. Nc c) /\ (x i^i y) = (/)) -> E.p e. NC Nc a = Tc p))
43	42	3adant1 973	. . . . . . . . . . . . . . . 16 |- ((b = (n u. m) /\ x = ~P1 n /\ y = ~P1 m) -> (((x e. Nc a /\ y e. Nc c) /\ (x i^i y) = (/)) -> E.p e. NC Nc a = Tc p))
44	43	exlimivv 1635	. . . . . . . . . . . . . . 15 |- (E.nE.m(b = (n u. m) /\ x = ~P1 n /\ y = ~P1 m) -> (((x e. Nc a /\ y e. Nc c) /\ (x i^i y) = (/)) -> E.p e. NC Nc a = Tc p))
45	44	com12 27	. . . . . . . . . . . . . 14 |- (((x e. Nc a /\ y e. Nc c) /\ (x i^i y) = (/)) -> (E.nE.m(b = (n u. m) /\ x = ~P1 n /\ y = ~P1 m) -> E.p e. NC Nc a = Tc p))
46	23, 45	syl5bi 208	. . . . . . . . . . . . 13 |- (((x e. Nc a /\ y e. Nc c) /\ (x i^i y) = (/)) -> (~P1 b = (x u. y) -> E.p e. NC Nc a = Tc p))
47	46	expimpd 586	. . . . . . . . . . . 12 |- ((x e. Nc a /\ y e. Nc c) -> (((x i^i y) = (/) /\ ~P1 b = (x u. y)) -> E.p e. NC Nc a = Tc p))
48	47	rexlimivv 2744	. . . . . . . . . . 11 |- (E.x e. Nc aE.y e. Nc c((x i^i y) = (/) /\ ~P1 b = (x u. y)) -> E.p e. NC Nc a = Tc p)
49	20, 48	sylbi 187	. . . . . . . . . 10 |- (~P1 b e. ( Nc a +o Nc c) -> E.p e. NC Nc a = Tc p)
50	19, 49	sylbi 187	. . . . . . . . 9 |- ( Nc ~P1 b = ( Nc a +o Nc c) -> E.p e. NC Nc a = Tc p)
51		tceq 6159	. . . . . . . . . . . . 13 |- (N = Nc b -> Tc N = Tc Nc b)
52		vex 2863	. . . . . . . . . . . . . 14 |- b e. _V
53	52	tcnc 6226	. . . . . . . . . . . . 13 |- Tc Nc b = Nc ~P1 b
54	51, 53	syl6eq 2401	. . . . . . . . . . . 12 |- (N = Nc b -> Tc N = Nc ~P1 b)
55	54	3ad2ant2 977	. . . . . . . . . . 11 |- ((M = Nc a /\ N = Nc b /\ q = Nc c) -> Tc N = Nc ~P1 b)
56		addceq12 4386	. . . . . . . . . . . 12 |- ((M = Nc a /\ q = Nc c) -> (M +o q) = ( Nc a +o Nc c))
57	56	3adant2 974	. . . . . . . . . . 11 |- ((M = Nc a /\ N = Nc b /\ q = Nc c) -> (M +o q) = ( Nc a +o Nc c))
58	55, 57	eqeq12d 2367	. . . . . . . . . 10 |- ((M = Nc a /\ N = Nc b /\ q = Nc c) -> ( Tc N = (M +o q) <-> Nc ~P1 b = ( Nc a +o Nc c)))
59		eqeq1 2359	. . . . . . . . . . . 12 |- (M = Nc a -> (M = Tc p <-> Nc a = Tc p))
60	59	rexbidv 2636	. . . . . . . . . . 11 |- (M = Nc a -> (E.p e. NC M = Tc p <-> E.p e. NC Nc a = Tc p))
61	60	3ad2ant1 976	. . . . . . . . . 10 |- ((M = Nc a /\ N = Nc b /\ q = Nc c) -> (E.p e. NC M = Tc p <-> E.p e. NC Nc a = Tc p))
62	58, 61	imbi12d 311	. . . . . . . . 9 |- ((M = Nc a /\ N = Nc b /\ q = Nc c) -> (( Tc N = (M +o q) -> E.p e. NC M = Tc p) <-> ( Nc ~P1 b = ( Nc a +o Nc c) -> E.p e. NC Nc a = Tc p)))
63	50, 62	mpbiri 224	. . . . . . . 8 |- ((M = Nc a /\ N = Nc b /\ q = Nc c) -> ( Tc N = (M +o q) -> E.p e. NC M = Tc p))
64	63	exlimiv 1634	. . . . . . 7 |- (E.c(M = Nc a /\ N = Nc b /\ q = Nc c) -> ( Tc N = (M +o q) -> E.p e. NC M = Tc p))
65	64	exlimivv 1635	. . . . . 6 |- (E.aE.bE.c(M = Nc a /\ N = Nc b /\ q = Nc c) -> ( Tc N = (M +o q) -> E.p e. NC M = Tc p))
66	9, 65	sylbi 187	. . . . 5 |- ((M e. NC /\ N e. NC /\ q e. NC ) -> ( Tc N = (M +o q) -> E.p e. NC M = Tc p))
67	66	3expa 1151	. . . 4 |- (((M e. NC /\ N e. NC ) /\ q e. NC ) -> ( Tc N = (M +o q) -> E.p e. NC M = Tc p))
68	67	rexlimdva 2739	. . 3 |- ((M e. NC /\ N e. NC ) -> (E.q e. NC Tc N = (M +o q) -> E.p e. NC M = Tc p))
69	3, 68	sylbid 206	. 2 |- ((M e. NC /\ N e. NC ) -> (M <_c Tc N -> E.p e. NC M = Tc p))
70	69	3impia 1148	1 |- ((M e. NC /\ N e. NC /\ M <_c Tc N) -> E.p e. NC M = Tc p)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616   u. cun 3208   i^i cin 3209  (/)c0 3551  ~P₁ cpw1 4136   +o cplc 4376   class class class wbr 4640   NC cncs 6089   <__c clec 6090   Nc cnc 6092   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  nchoicelem19  6308