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 ∈ NC → ∃!x ∈ NC ∃y ∈ A x = Nc ℘1y)

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

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617  ∅c0 3551  ℘₁cpw1 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