Theorem taddc 6230

Description: T raising rule for cardinal sum. (Contributed by SF, 11-Mar-2015.)

Assertion

Ref	Expression
taddc	⊢ (((A ∈ NC ∧ B ∈ NC ∧ X ∈ NC ) ∧ Tc A = ( Tc B +c X)) → ∃c ∈ NC X = Tc c)

Distinct variable group:   X,c

Allowed substitution hints:   A(c)   B(c)

Proof of Theorem taddc

Dummy variables a b w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elncs 6120	. . . . . 6 ⊢ (A ∈ NC ↔ ∃x A = Nc x)
2		elncs 6120	. . . . . 6 ⊢ (B ∈ NC ↔ ∃y B = Nc y)
3		elncs 6120	. . . . . 6 ⊢ (X ∈ NC ↔ ∃z X = Nc z)
4	1, 2, 3	3anbi123i 1140	. . . . 5 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ X ∈ NC ) ↔ (∃x A = Nc x ∧ ∃y B = Nc y ∧ ∃z X = Nc z))
5		eeeanv 1914	. . . . 5 ⊢ (∃x∃y∃z(A = Nc x ∧ B = Nc y ∧ X = Nc z) ↔ (∃x A = Nc x ∧ ∃y B = Nc y ∧ ∃z X = Nc z))
6	4, 5	bitr4i 243	. . . 4 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ X ∈ NC ) ↔ ∃x∃y∃z(A = Nc x ∧ B = Nc y ∧ X = Nc z))
7		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
8	7	tcnc 6226	. . . . . . . . . 10 ⊢ Tc Nc x = Nc ℘1x
9		vex 2863	. . . . . . . . . . . 12 ⊢ y ∈ V
10	9	tcnc 6226	. . . . . . . . . . 11 ⊢ Tc Nc y = Nc ℘1y
11	10	addceq1i 4387	. . . . . . . . . 10 ⊢ ( Tc Nc y +c Nc z) = ( Nc ℘1y +c Nc z)
12	8, 11	eqeq12i 2366	. . . . . . . . 9 ⊢ ( Tc Nc x = ( Tc Nc y +c Nc z) ↔ Nc ℘1x = ( Nc ℘1y +c Nc z))
13		eqcom 2355	. . . . . . . . 9 ⊢ ( Nc ℘1x = ( Nc ℘1y +c Nc z) ↔ ( Nc ℘1y +c Nc z) = Nc ℘1x)
14	9	pw1ex 4304	. . . . . . . . . . . 12 ⊢ ℘1y ∈ V
15	14	ncelncsi 6122	. . . . . . . . . . 11 ⊢ Nc ℘1y ∈ NC
16		vex 2863	. . . . . . . . . . . 12 ⊢ z ∈ V
17	16	ncelncsi 6122	. . . . . . . . . . 11 ⊢ Nc z ∈ NC
18		ncaddccl 6145	. . . . . . . . . . 11 ⊢ (( Nc ℘1y ∈ NC ∧ Nc z ∈ NC ) → ( Nc ℘1y +c Nc z) ∈ NC )
19	15, 17, 18	mp2an 653	. . . . . . . . . 10 ⊢ ( Nc ℘1y +c Nc z) ∈ NC
20		ncseqnc 6129	. . . . . . . . . 10 ⊢ (( Nc ℘1y +c Nc z) ∈ NC → (( Nc ℘1y +c Nc z) = Nc ℘1x ↔ ℘1x ∈ ( Nc ℘1y +c Nc z)))
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ (( Nc ℘1y +c Nc z) = Nc ℘1x ↔ ℘1x ∈ ( Nc ℘1y +c Nc z))
22	12, 13, 21	3bitri 262	. . . . . . . 8 ⊢ ( Tc Nc x = ( Tc Nc y +c Nc z) ↔ ℘1x ∈ ( Nc ℘1y +c Nc z))
23		eladdc 4399	. . . . . . . . 9 ⊢ (℘1x ∈ ( Nc ℘1y +c Nc z) ↔ ∃a ∈ Nc ℘1y∃b ∈ Nc z((a ∩ b) = ∅ ∧ ℘1x = (a ∪ b)))
24		vex 2863	. . . . . . . . . . . . . 14 ⊢ a ∈ V
25		vex 2863	. . . . . . . . . . . . . 14 ⊢ b ∈ V
26	24, 25	pw1equn 4332	. . . . . . . . . . . . 13 ⊢ (℘1x = (a ∪ b) ↔ ∃c∃w(x = (c ∪ w) ∧ a = ℘1c ∧ b = ℘1w))
27		simp3 957	. . . . . . . . . . . . . . . 16 ⊢ ((x = (c ∪ w) ∧ a = ℘1c ∧ b = ℘1w) → b = ℘1w)
28		elnc 6126	. . . . . . . . . . . . . . . . 17 ⊢ (b ∈ Nc z ↔ b ≈ z)
29		ensym 6038	. . . . . . . . . . . . . . . . . 18 ⊢ (b ≈ z ↔ z ≈ b)
30		breq2 4644	. . . . . . . . . . . . . . . . . . 19 ⊢ (b = ℘1w → (z ≈ b ↔ z ≈ ℘1w))
31	30	biimpcd 215	. . . . . . . . . . . . . . . . . 18 ⊢ (z ≈ b → (b = ℘1w → z ≈ ℘1w))
32	29, 31	sylbi 187	. . . . . . . . . . . . . . . . 17 ⊢ (b ≈ z → (b = ℘1w → z ≈ ℘1w))
33	28, 32	sylbi 187	. . . . . . . . . . . . . . . 16 ⊢ (b ∈ Nc z → (b = ℘1w → z ≈ ℘1w))
34	27, 33	syl5 28	. . . . . . . . . . . . . . 15 ⊢ (b ∈ Nc z → ((x = (c ∪ w) ∧ a = ℘1c ∧ b = ℘1w) → z ≈ ℘1w))
35	34	eximdv 1622	. . . . . . . . . . . . . 14 ⊢ (b ∈ Nc z → (∃w(x = (c ∪ w) ∧ a = ℘1c ∧ b = ℘1w) → ∃w z ≈ ℘1w))
36	35	exlimdv 1636	. . . . . . . . . . . . 13 ⊢ (b ∈ Nc z → (∃c∃w(x = (c ∪ w) ∧ a = ℘1c ∧ b = ℘1w) → ∃w z ≈ ℘1w))
37	26, 36	syl5bi 208	. . . . . . . . . . . 12 ⊢ (b ∈ Nc z → (℘1x = (a ∪ b) → ∃w z ≈ ℘1w))
38	37	adantld 453	. . . . . . . . . . 11 ⊢ (b ∈ Nc z → (((a ∩ b) = ∅ ∧ ℘1x = (a ∪ b)) → ∃w z ≈ ℘1w))
39	38	rexlimiv 2733	. . . . . . . . . 10 ⊢ (∃b ∈ Nc z((a ∩ b) = ∅ ∧ ℘1x = (a ∪ b)) → ∃w z ≈ ℘1w)
40	39	rexlimivw 2735	. . . . . . . . 9 ⊢ (∃a ∈ Nc ℘1y∃b ∈ Nc z((a ∩ b) = ∅ ∧ ℘1x = (a ∪ b)) → ∃w z ≈ ℘1w)
41	23, 40	sylbi 187	. . . . . . . 8 ⊢ (℘1x ∈ ( Nc ℘1y +c Nc z) → ∃w z ≈ ℘1w)
42	22, 41	sylbi 187	. . . . . . 7 ⊢ ( Tc Nc x = ( Tc Nc y +c Nc z) → ∃w z ≈ ℘1w)
43		tceq 6159	. . . . . . . . . 10 ⊢ (A = Nc x → Tc A = Tc Nc x)
44	43	3ad2ant1 976	. . . . . . . . 9 ⊢ ((A = Nc x ∧ B = Nc y ∧ X = Nc z) → Tc A = Tc Nc x)
45		tceq 6159	. . . . . . . . . . . 12 ⊢ (B = Nc y → Tc B = Tc Nc y)
46	45	adantr 451	. . . . . . . . . . 11 ⊢ ((B = Nc y ∧ X = Nc z) → Tc B = Tc Nc y)
47		simpr 447	. . . . . . . . . . 11 ⊢ ((B = Nc y ∧ X = Nc z) → X = Nc z)
48	46, 47	addceq12d 4392	. . . . . . . . . 10 ⊢ ((B = Nc y ∧ X = Nc z) → ( Tc B +c X) = ( Tc Nc y +c Nc z))
49	48	3adant1 973	. . . . . . . . 9 ⊢ ((A = Nc x ∧ B = Nc y ∧ X = Nc z) → ( Tc B +c X) = ( Tc Nc y +c Nc z))
50	44, 49	eqeq12d 2367	. . . . . . . 8 ⊢ ((A = Nc x ∧ B = Nc y ∧ X = Nc z) → ( Tc A = ( Tc B +c X) ↔ Tc Nc x = ( Tc Nc y +c Nc z)))
51		eqeq1 2359	. . . . . . . . . . 11 ⊢ (X = Nc z → (X = Nc ℘1w ↔ Nc z = Nc ℘1w))
52	16	eqnc 6128	. . . . . . . . . . 11 ⊢ ( Nc z = Nc ℘1w ↔ z ≈ ℘1w)
53	51, 52	syl6bb 252	. . . . . . . . . 10 ⊢ (X = Nc z → (X = Nc ℘1w ↔ z ≈ ℘1w))
54	53	exbidv 1626	. . . . . . . . 9 ⊢ (X = Nc z → (∃w X = Nc ℘1w ↔ ∃w z ≈ ℘1w))
55	54	3ad2ant3 978	. . . . . . . 8 ⊢ ((A = Nc x ∧ B = Nc y ∧ X = Nc z) → (∃w X = Nc ℘1w ↔ ∃w z ≈ ℘1w))
56	50, 55	imbi12d 311	. . . . . . 7 ⊢ ((A = Nc x ∧ B = Nc y ∧ X = Nc z) → (( Tc A = ( Tc B +c X) → ∃w X = Nc ℘1w) ↔ ( Tc Nc x = ( Tc Nc y +c Nc z) → ∃w z ≈ ℘1w)))
57	42, 56	mpbiri 224	. . . . . 6 ⊢ ((A = Nc x ∧ B = Nc y ∧ X = Nc z) → ( Tc A = ( Tc B +c X) → ∃w X = Nc ℘1w))
58	57	exlimiv 1634	. . . . 5 ⊢ (∃z(A = Nc x ∧ B = Nc y ∧ X = Nc z) → ( Tc A = ( Tc B +c X) → ∃w X = Nc ℘1w))
59	58	exlimivv 1635	. . . 4 ⊢ (∃x∃y∃z(A = Nc x ∧ B = Nc y ∧ X = Nc z) → ( Tc A = ( Tc B +c X) → ∃w X = Nc ℘1w))
60	6, 59	sylbi 187	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ X ∈ NC ) → ( Tc A = ( Tc B +c X) → ∃w X = Nc ℘1w))
61	60	imp 418	. 2 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ X ∈ NC ) ∧ Tc A = ( Tc B +c X)) → ∃w X = Nc ℘1w)
62		df-rex 2621	. . 3 ⊢ (∃c ∈ NC X = Tc c ↔ ∃c(c ∈ NC ∧ X = Tc c))
63		elncs 6120	. . . . . 6 ⊢ (c ∈ NC ↔ ∃w c = Nc w)
64	63	anbi1i 676	. . . . 5 ⊢ ((c ∈ NC ∧ X = Tc c) ↔ (∃w c = Nc w ∧ X = Tc c))
65		19.41v 1901	. . . . 5 ⊢ (∃w(c = Nc w ∧ X = Tc c) ↔ (∃w c = Nc w ∧ X = Tc c))
66	64, 65	bitr4i 243	. . . 4 ⊢ ((c ∈ NC ∧ X = Tc c) ↔ ∃w(c = Nc w ∧ X = Tc c))
67	66	exbii 1582	. . 3 ⊢ (∃c(c ∈ NC ∧ X = Tc c) ↔ ∃c∃w(c = Nc w ∧ X = Tc c))
68		excom 1741	. . . 4 ⊢ (∃c∃w(c = Nc w ∧ X = Tc c) ↔ ∃w∃c(c = Nc w ∧ X = Tc c))
69		ncex 6118	. . . . . 6 ⊢ Nc w ∈ V
70		tceq 6159	. . . . . . . 8 ⊢ (c = Nc w → Tc c = Tc Nc w)
71		vex 2863	. . . . . . . . 9 ⊢ w ∈ V
72	71	tcnc 6226	. . . . . . . 8 ⊢ Tc Nc w = Nc ℘1w
73	70, 72	syl6eq 2401	. . . . . . 7 ⊢ (c = Nc w → Tc c = Nc ℘1w)
74	73	eqeq2d 2364	. . . . . 6 ⊢ (c = Nc w → (X = Tc c ↔ X = Nc ℘1w))
75	69, 74	ceqsexv 2895	. . . . 5 ⊢ (∃c(c = Nc w ∧ X = Tc c) ↔ X = Nc ℘1w)
76	75	exbii 1582	. . . 4 ⊢ (∃w∃c(c = Nc w ∧ X = Tc c) ↔ ∃w X = Nc ℘1w)
77	68, 76	bitri 240	. . 3 ⊢ (∃c∃w(c = Nc w ∧ X = Tc c) ↔ ∃w X = Nc ℘1w)
78	62, 67, 77	3bitri 262	. 2 ⊢ (∃c ∈ NC X = Tc c ↔ ∃w X = Nc ℘1w)
79	61, 78	sylibr 203	1 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ X ∈ NC ) ∧ Tc A = ( Tc B +c X)) → ∃c ∈ NC X = Tc c)

Syntax hints:   → 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   ≈ cen 6029   NC cncs 6089   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-nc 6102  df-tc 6104

This theorem is referenced by:  tlecg  6231