Theorem ncspw1eu 6160

Description: Given a cardinal, there is a unique cardinal that contains the unit power class of its members. (Contributed by SF, 2-Mar-2015.)

Assertion

Ref	Expression
ncspw1eu	|- (A e. NC -> E!x e. NC E.y e. A x = Nc ~P1 y)

Distinct variable group:   x,A,y

Proof of Theorem ncspw1eu

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nulnnc 6119	. . . . . . 7 |- -. (/) e. NC
2		eleq1 2413	. . . . . . 7 |- (A = (/) -> (A e. NC <-> (/) e. NC ))
3	1, 2	mtbiri 294	. . . . . 6 |- (A = (/) -> -. A e. NC )
4	3	necon2ai 2562	. . . . 5 |- (A e. NC -> A =/= (/))
5		n0 3560	. . . . 5 |- (A =/= (/) <-> E.y y e. A)
6	4, 5	sylib 188	. . . 4 |- (A e. NC -> E.y y e. A)
7		vex 2863	. . . . . . . . . 10 |- y e. _V
8	7	pw1ex 4304	. . . . . . . . 9 |- ~P1 y e. _V
9	8	ncelncsi 6122	. . . . . . . 8 |- Nc ~P1 y e. NC
10		eqid 2353	. . . . . . . 8 |- Nc ~P1 y = Nc ~P1 y
11		eqeq1 2359	. . . . . . . . 9 |- (x = Nc ~P1 y -> (x = Nc ~P1 y <-> Nc ~P1 y = Nc ~P1 y))
12	11	rspcev 2956	. . . . . . . 8 |- (( Nc ~P1 y e. NC /\ Nc ~P1 y = Nc ~P1 y) -> E.x e. NC x = Nc ~P1 y)
13	9, 10, 12	mp2an 653	. . . . . . 7 |- E.x e. NC x = Nc ~P1 y
14	13	jctr 526	. . . . . 6 |- (y e. A -> (y e. A /\ E.x e. NC x = Nc ~P1 y))
15	14	a1i 10	. . . . 5 |- (A e. NC -> (y e. A -> (y e. A /\ E.x e. NC x = Nc ~P1 y)))
16	15	eximdv 1622	. . . 4 |- (A e. NC -> (E.y y e. A -> E.y(y e. A /\ E.x e. NC x = Nc ~P1 y)))
17	6, 16	mpd 14	. . 3 |- (A e. NC -> E.y(y e. A /\ E.x e. NC x = Nc ~P1 y))
18		rexcom 2773	. . . 4 |- (E.x e. NC E.y e. A x = Nc ~P1 y <-> E.y e. A E.x e. NC x = Nc ~P1 y)
19		df-rex 2621	. . . 4 |- (E.y e. A E.x e. NC x = Nc ~P1 y <-> E.y(y e. A /\ E.x e. NC x = Nc ~P1 y))
20	18, 19	bitri 240	. . 3 |- (E.x e. NC E.y e. A x = Nc ~P1 y <-> E.y(y e. A /\ E.x e. NC x = Nc ~P1 y))
21	17, 20	sylibr 203	. 2 |- (A e. NC -> E.x e. NC E.y e. A x = Nc ~P1 y)
22		reeanv 2779	. . . 4 |- (E.y e. A E.w e. A (x = Nc ~P1 y /\ z = Nc ~P1 w) <-> (E.y e. A x = Nc ~P1 y /\ E.w e. A z = Nc ~P1 w))
23		ncseqnc 6129	. . . . . . . . . . . 12 |- (A e. NC -> (A = Nc y <-> y e. A))
24	23	biimpar 471	. . . . . . . . . . 11 |- ((A e. NC /\ y e. A) -> A = Nc y)
25	24	adantrr 697	. . . . . . . . . 10 |- ((A e. NC /\ (y e. A /\ w e. A)) -> A = Nc y)
26		ncseqnc 6129	. . . . . . . . . . . 12 |- (A e. NC -> (A = Nc w <-> w e. A))
27	26	biimpar 471	. . . . . . . . . . 11 |- ((A e. NC /\ w e. A) -> A = Nc w)
28	27	adantrl 696	. . . . . . . . . 10 |- ((A e. NC /\ (y e. A /\ w e. A)) -> A = Nc w)
29	25, 28	eqtr3d 2387	. . . . . . . . 9 |- ((A e. NC /\ (y e. A /\ w e. A)) -> Nc y = Nc w)
30	7	ncpw1 6153	. . . . . . . . 9 |- ( Nc y = Nc w <-> Nc ~P1 y = Nc ~P1 w)
31	29, 30	sylib 188	. . . . . . . 8 |- ((A e. NC /\ (y e. A /\ w e. A)) -> Nc ~P1 y = Nc ~P1 w)
32	31	3adant2 974	. . . . . . 7 |- ((A e. NC /\ (x e. NC /\ z e. NC ) /\ (y e. A /\ w e. A)) -> Nc ~P1 y = Nc ~P1 w)
33		eqeq2 2362	. . . . . . . . 9 |- ( Nc ~P1 y = Nc ~P1 w -> (x = Nc ~P1 y <-> x = Nc ~P1 w))
34	33	anbi1d 685	. . . . . . . 8 |- ( Nc ~P1 y = Nc ~P1 w -> ((x = Nc ~P1 y /\ z = Nc ~P1 w) <-> (x = Nc ~P1 w /\ z = Nc ~P1 w)))
35		eqtr3 2372	. . . . . . . 8 |- ((x = Nc ~P1 w /\ z = Nc ~P1 w) -> x = z)
36	34, 35	syl6bi 219	. . . . . . 7 |- ( Nc ~P1 y = Nc ~P1 w -> ((x = Nc ~P1 y /\ z = Nc ~P1 w) -> x = z))
37	32, 36	syl 15	. . . . . 6 |- ((A e. NC /\ (x e. NC /\ z e. NC ) /\ (y e. A /\ w e. A)) -> ((x = Nc ~P1 y /\ z = Nc ~P1 w) -> x = z))
38	37	3expa 1151	. . . . 5 |- (((A e. NC /\ (x e. NC /\ z e. NC )) /\ (y e. A /\ w e. A)) -> ((x = Nc ~P1 y /\ z = Nc ~P1 w) -> x = z))
39	38	rexlimdvva 2746	. . . 4 |- ((A e. NC /\ (x e. NC /\ z e. NC )) -> (E.y e. A E.w e. A (x = Nc ~P1 y /\ z = Nc ~P1 w) -> x = z))
40	22, 39	syl5bir 209	. . 3 |- ((A e. NC /\ (x e. NC /\ z e. NC )) -> ((E.y e. A x = Nc ~P1 y /\ E.w e. A z = Nc ~P1 w) -> x = z))
41	40	ralrimivva 2707	. 2 |- (A e. NC -> A.x e. NC A.z e. NC ((E.y e. A x = Nc ~P1 y /\ E.w e. A z = Nc ~P1 w) -> x = z))
42		eqeq1 2359	. . . . 5 |- (x = z -> (x = Nc ~P1 y <-> z = Nc ~P1 y))
43	42	rexbidv 2636	. . . 4 |- (x = z -> (E.y e. A x = Nc ~P1 y <-> E.y e. A z = Nc ~P1 y))
44		pw1eq 4144	. . . . . . 7 |- (y = w -> ~P1 y = ~P1 w)
45	44	nceqd 6111	. . . . . 6 |- (y = w -> Nc ~P1 y = Nc ~P1 w)
46	45	eqeq2d 2364	. . . . 5 |- (y = w -> (z = Nc ~P1 y <-> z = Nc ~P1 w))
47	46	cbvrexv 2837	. . . 4 |- (E.y e. A z = Nc ~P1 y <-> E.w e. A z = Nc ~P1 w)
48	43, 47	syl6bb 252	. . 3 |- (x = z -> (E.y e. A x = Nc ~P1 y <-> E.w e. A z = Nc ~P1 w))
49	48	reu4 3031	. 2 |- (E!x e. NC E.y e. A x = Nc ~P1 y <-> (E.x e. NC E.y e. A x = Nc ~P1 y /\ A.x e. NC A.z e. NC ((E.y e. A x = Nc ~P1 y /\ E.w e. A z = Nc ~P1 w) -> x = z)))
50	21, 41, 49	sylanbrc 645	1 |- (A e. NC -> E!x e. NC E.y e. A x = Nc ~P1 y)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615  E.wrex 2616  E!wreu 2617  (/)c0 3551  ~P₁ cpw1 4136   NC cncs 6089   Nc cnc 6092

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

This theorem is referenced by:  tccl  6161  eqtc  6162