Theorem 1cnc 6140

Description: Cardinal one is a cardinal number. Corollary 2 to theorem XI.2.8 of [Rosser] p. 373. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
1cnc	⊢ 1c ∈ NC

Proof of Theorem 1cnc

Dummy variables x f y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfec2 5949	. . . 4 ⊢ [{y}] ≈ = {z ∣ {y} ≈ z}
2		df-nc 6102	. . . 4 ⊢ Nc {y} = [{y}] ≈
3		el1c 4140	. . . . . 6 ⊢ (z ∈ 1c ↔ ∃x z = {x})
4		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
5		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
6		en2sn 6048	. . . . . . . . . 10 ⊢ ((y ∈ V ∧ x ∈ V) → {y} ≈ {x})
7	4, 5, 6	mp2an 653	. . . . . . . . 9 ⊢ {y} ≈ {x}
8		breq2 4644	. . . . . . . . 9 ⊢ (z = {x} → ({y} ≈ z ↔ {y} ≈ {x}))
9	7, 8	mpbiri 224	. . . . . . . 8 ⊢ (z = {x} → {y} ≈ z)
10	9	exlimiv 1634	. . . . . . 7 ⊢ (∃x z = {x} → {y} ≈ z)
11		bren 6031	. . . . . . . 8 ⊢ ({y} ≈ z ↔ ∃f f:{y}–1-1-onto→z)
12		f1of 5288	. . . . . . . . . 10 ⊢ (f:{y}–1-1-onto→z → f:{y}–→z)
13		f1ofo 5294	. . . . . . . . . . 11 ⊢ (f:{y}–1-1-onto→z → f:{y}–onto→z)
14		forn 5273	. . . . . . . . . . 11 ⊢ (f:{y}–onto→z → ran f = z)
15	13, 14	syl 15	. . . . . . . . . 10 ⊢ (f:{y}–1-1-onto→z → ran f = z)
16	4	fsn2 5435	. . . . . . . . . . 11 ⊢ (f:{y}–→z ↔ ((f ‘y) ∈ z ∧ f = {⟨y, (f ‘y)⟩}))
17		rneq 4957	. . . . . . . . . . . . . . 15 ⊢ (f = {⟨y, (f ‘y)⟩} → ran f = ran {⟨y, (f ‘y)⟩})
18	4	rnsnop 5076	. . . . . . . . . . . . . . 15 ⊢ ran {⟨y, (f ‘y)⟩} = {(f ‘y)}
19	17, 18	syl6eq 2401	. . . . . . . . . . . . . 14 ⊢ (f = {⟨y, (f ‘y)⟩} → ran f = {(f ‘y)})
20	19	eqeq1d 2361	. . . . . . . . . . . . 13 ⊢ (f = {⟨y, (f ‘y)⟩} → (ran f = z ↔ {(f ‘y)} = z))
21		fvex 5340	. . . . . . . . . . . . . . 15 ⊢ (f ‘y) ∈ V
22		sneq 3745	. . . . . . . . . . . . . . . 16 ⊢ (x = (f ‘y) → {x} = {(f ‘y)})
23	22	eqeq2d 2364	. . . . . . . . . . . . . . 15 ⊢ (x = (f ‘y) → (z = {x} ↔ z = {(f ‘y)}))
24	21, 23	spcev 2947	. . . . . . . . . . . . . 14 ⊢ (z = {(f ‘y)} → ∃x z = {x})
25	24	eqcoms 2356	. . . . . . . . . . . . 13 ⊢ ({(f ‘y)} = z → ∃x z = {x})
26	20, 25	syl6bi 219	. . . . . . . . . . . 12 ⊢ (f = {⟨y, (f ‘y)⟩} → (ran f = z → ∃x z = {x}))
27	26	adantl 452	. . . . . . . . . . 11 ⊢ (((f ‘y) ∈ z ∧ f = {⟨y, (f ‘y)⟩}) → (ran f = z → ∃x z = {x}))
28	16, 27	sylbi 187	. . . . . . . . . 10 ⊢ (f:{y}–→z → (ran f = z → ∃x z = {x}))
29	12, 15, 28	sylc 56	. . . . . . . . 9 ⊢ (f:{y}–1-1-onto→z → ∃x z = {x})
30	29	exlimiv 1634	. . . . . . . 8 ⊢ (∃f f:{y}–1-1-onto→z → ∃x z = {x})
31	11, 30	sylbi 187	. . . . . . 7 ⊢ ({y} ≈ z → ∃x z = {x})
32	10, 31	impbii 180	. . . . . 6 ⊢ (∃x z = {x} ↔ {y} ≈ z)
33	3, 32	bitri 240	. . . . 5 ⊢ (z ∈ 1c ↔ {y} ≈ z)
34	33	abbi2i 2465	. . . 4 ⊢ 1c = {z ∣ {y} ≈ z}
35	1, 2, 34	3eqtr4ri 2384	. . 3 ⊢ 1c = Nc {y}
36		snex 4112	. . . 4 ⊢ {y} ∈ V
37		nceq 6109	. . . . 5 ⊢ (x = {y} → Nc x = Nc {y})
38	37	eqeq2d 2364	. . . 4 ⊢ (x = {y} → (1c = Nc x ↔ 1c = Nc {y}))
39	36, 38	spcev 2947	. . 3 ⊢ (1c = Nc {y} → ∃x1c = Nc x)
40	35, 39	ax-mp 5	. 2 ⊢ ∃x1c = Nc x
41		elncs 6120	. 2 ⊢ (1c ∈ NC ↔ ∃x1c = Nc x)
42	40, 41	mpbir 200	1 ⊢ 1c ∈ NC

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860  {csn 3738  1_cc1c 4135  ⟨cop 4562   class class class wbr 4640  ran crn 4774  –→wf 4778  –onto→wfo 4780  –1-1-onto→wf1o 4781   ‘cfv 4782  [cec 5946   ≈ cen 6029   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-fv 4796  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-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102

This theorem is referenced by:  df1c3  6141  peano2nc  6146  tc1c  6166  tc2c  6167  ce0nn  6181  nc0suc  6218  leconnnc  6219  ncslemuc  6256  nnltp1c  6263  ncslesuc  6268  nmembers1lem2  6270  nmembers1lem3  6271  nmembers1  6272  nnc3n3p1  6279  nchoicelem1  6290  nchoicelem2  6291  nchoicelem13  6302  nchoicelem14  6303  nchoicelem17  6306