Theorem dfnnc3 5886

Description: The finite cardinals as expressed via the closure operation. Theorem X.1.3 of [Rosser] p. 276. (Contributed by SF, 12-Feb-2015.)

Assertion

Ref	Expression
dfnnc3	⊢ Nn = Clos1 ({0c}, (x ∈ V ↦ (x +c 1c)))

Proof of Theorem dfnnc3

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

Step	Hyp	Ref	Expression
1		0cex 4393	. . . . . 6 ⊢ 0c ∈ V
2	1	snss 3839	. . . . 5 ⊢ (0c ∈ a ↔ {0c} ⊆ a)
3		dfss2 3263	. . . . . 6 ⊢ (((x ∈ V ↦ (x +c 1c)) “ a) ⊆ a ↔ ∀z(z ∈ ((x ∈ V ↦ (x +c 1c)) “ a) → z ∈ a))
4		ralcom4 2878	. . . . . . 7 ⊢ (∀y ∈ a ∀z(((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a) ↔ ∀z∀y ∈ a (((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a))
5		eqid 2353	. . . . . . . . . . . . 13 ⊢ (x ∈ V ↦ (x +c 1c)) = (x ∈ V ↦ (x +c 1c))
6	5	fnmpt 5690	. . . . . . . . . . . 12 ⊢ (∀x ∈ V (x +c 1c) ∈ V → (x ∈ V ↦ (x +c 1c)) Fn V)
7		vex 2863	. . . . . . . . . . . . . 14 ⊢ x ∈ V
8		1cex 4143	. . . . . . . . . . . . . 14 ⊢ 1c ∈ V
9	7, 8	addcex 4395	. . . . . . . . . . . . 13 ⊢ (x +c 1c) ∈ V
10	9	a1i 10	. . . . . . . . . . . 12 ⊢ (x ∈ V → (x +c 1c) ∈ V)
11	6, 10	mprg 2684	. . . . . . . . . . 11 ⊢ (x ∈ V ↦ (x +c 1c)) Fn V
12		ssv 3292	. . . . . . . . . . 11 ⊢ a ⊆ V
13		fvelimab 5371	. . . . . . . . . . 11 ⊢ (((x ∈ V ↦ (x +c 1c)) Fn V ∧ a ⊆ V) → (z ∈ ((x ∈ V ↦ (x +c 1c)) “ a) ↔ ∃y ∈ a ((x ∈ V ↦ (x +c 1c)) ‘y) = z))
14	11, 12, 13	mp2an 653	. . . . . . . . . 10 ⊢ (z ∈ ((x ∈ V ↦ (x +c 1c)) “ a) ↔ ∃y ∈ a ((x ∈ V ↦ (x +c 1c)) ‘y) = z)
15	14	imbi1i 315	. . . . . . . . 9 ⊢ ((z ∈ ((x ∈ V ↦ (x +c 1c)) “ a) → z ∈ a) ↔ (∃y ∈ a ((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a))
16		r19.23v 2731	. . . . . . . . 9 ⊢ (∀y ∈ a (((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a) ↔ (∃y ∈ a ((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a))
17	15, 16	bitr4i 243	. . . . . . . 8 ⊢ ((z ∈ ((x ∈ V ↦ (x +c 1c)) “ a) → z ∈ a) ↔ ∀y ∈ a (((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a))
18	17	albii 1566	. . . . . . 7 ⊢ (∀z(z ∈ ((x ∈ V ↦ (x +c 1c)) “ a) → z ∈ a) ↔ ∀z∀y ∈ a (((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a))
19	4, 18	bitr4i 243	. . . . . 6 ⊢ (∀y ∈ a ∀z(((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a) ↔ ∀z(z ∈ ((x ∈ V ↦ (x +c 1c)) “ a) → z ∈ a))
20		vex 2863	. . . . . . . . . . . . 13 ⊢ y ∈ V
21		addceq1 4384	. . . . . . . . . . . . . 14 ⊢ (x = y → (x +c 1c) = (y +c 1c))
22	20, 8	addcex 4395	. . . . . . . . . . . . . 14 ⊢ (y +c 1c) ∈ V
23	21, 5, 22	fvmpt 5701	. . . . . . . . . . . . 13 ⊢ (y ∈ V → ((x ∈ V ↦ (x +c 1c)) ‘y) = (y +c 1c))
24	20, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((x ∈ V ↦ (x +c 1c)) ‘y) = (y +c 1c)
25	24	eqeq1i 2360	. . . . . . . . . . 11 ⊢ (((x ∈ V ↦ (x +c 1c)) ‘y) = z ↔ (y +c 1c) = z)
26		eqcom 2355	. . . . . . . . . . 11 ⊢ ((y +c 1c) = z ↔ z = (y +c 1c))
27	25, 26	bitri 240	. . . . . . . . . 10 ⊢ (((x ∈ V ↦ (x +c 1c)) ‘y) = z ↔ z = (y +c 1c))
28	27	imbi1i 315	. . . . . . . . 9 ⊢ ((((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a) ↔ (z = (y +c 1c) → z ∈ a))
29	28	albii 1566	. . . . . . . 8 ⊢ (∀z(((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a) ↔ ∀z(z = (y +c 1c) → z ∈ a))
30		eleq1 2413	. . . . . . . . 9 ⊢ (z = (y +c 1c) → (z ∈ a ↔ (y +c 1c) ∈ a))
31	22, 30	ceqsalv 2886	. . . . . . . 8 ⊢ (∀z(z = (y +c 1c) → z ∈ a) ↔ (y +c 1c) ∈ a)
32	29, 31	bitri 240	. . . . . . 7 ⊢ (∀z(((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a) ↔ (y +c 1c) ∈ a)
33	32	ralbii 2639	. . . . . 6 ⊢ (∀y ∈ a ∀z(((x ∈ V ↦ (x +c 1c)) ‘y) = z → z ∈ a) ↔ ∀y ∈ a (y +c 1c) ∈ a)
34	3, 19, 33	3bitr2ri 265	. . . . 5 ⊢ (∀y ∈ a (y +c 1c) ∈ a ↔ ((x ∈ V ↦ (x +c 1c)) “ a) ⊆ a)
35	2, 34	anbi12i 678	. . . 4 ⊢ ((0c ∈ a ∧ ∀y ∈ a (y +c 1c) ∈ a) ↔ ({0c} ⊆ a ∧ ((x ∈ V ↦ (x +c 1c)) “ a) ⊆ a))
36	35	abbii 2466	. . 3 ⊢ {a ∣ (0c ∈ a ∧ ∀y ∈ a (y +c 1c) ∈ a)} = {a ∣ ({0c} ⊆ a ∧ ((x ∈ V ↦ (x +c 1c)) “ a) ⊆ a)}
37	36	inteqi 3931	. 2 ⊢ ∩{a ∣ (0c ∈ a ∧ ∀y ∈ a (y +c 1c) ∈ a)} = ∩{a ∣ ({0c} ⊆ a ∧ ((x ∈ V ↦ (x +c 1c)) “ a) ⊆ a)}
38		df-nnc 4380	. 2 ⊢ Nn = ∩{a ∣ (0c ∈ a ∧ ∀y ∈ a (y +c 1c) ∈ a)}
39		df-clos1 5874	. 2 ⊢ Clos1 ({0c}, (x ∈ V ↦ (x +c 1c))) = ∩{a ∣ ({0c} ⊆ a ∧ ((x ∈ V ↦ (x +c 1c)) “ a) ⊆ a)}
40	37, 38, 39	3eqtr4i 2383	1 ⊢ Nn = Clos1 ({0c}, (x ∈ V ↦ (x +c 1c)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  {csn 3738  ∩cint 3927  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   “ cima 4723   Fn wfn 4777   ‘cfv 4782   ↦ cmpt 5652   Clos1 cclos1 5873

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-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796  df-mpt 5653  df-clos1 5874

This theorem is referenced by: (None)