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

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  ℘₁cpw1 4136   +_c cplc 4376   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090   Nc cnc 6092   T_c ctc 6094

This proof depends on 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 proof 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 used by:  ce2le  6234  nchoicelem19  6308