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 e. _V |-> (x +o 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 e. _V
2	1	snss 3839	. . . . 5 |- (0c e. a <-> {0c} C_ a)
3		dfss2 3263	. . . . . 6 |- (((x e. _V |-> (x +o 1c))"a) C_ a <-> A.z(z e. ((x e. _V |-> (x +o 1c))"a) -> z e. a))
4		ralcom4 2878	. . . . . . 7 |- (A.y e. a A.z(((x e. _V |-> (x +o 1c))` y) = z -> z e. a) <-> A.zA.y e. a (((x e. _V |-> (x +o 1c))` y) = z -> z e. a))
5		eqid 2353	. . . . . . . . . . . . 13 |- (x e. _V |-> (x +o 1c)) = (x e. _V |-> (x +o 1c))
6	5	fnmpt 5690	. . . . . . . . . . . 12 |- (A.x e. _V (x +o 1c) e. _V -> (x e. _V |-> (x +o 1c)) Fn _V)
7		vex 2863	. . . . . . . . . . . . . 14 |- x e. _V
8		1cex 4143	. . . . . . . . . . . . . 14 |- 1c e. _V
9	7, 8	addcex 4395	. . . . . . . . . . . . 13 |- (x +o 1c) e. _V
10	9	a1i 10	. . . . . . . . . . . 12 |- (x e. _V -> (x +o 1c) e. _V)
11	6, 10	mprg 2684	. . . . . . . . . . 11 |- (x e. _V |-> (x +o 1c)) Fn _V
12		ssv 3292	. . . . . . . . . . 11 |- a C_ _V
13		fvelimab 5371	. . . . . . . . . . 11 |- (((x e. _V |-> (x +o 1c)) Fn _V /\ a C_ _V) -> (z e. ((x e. _V |-> (x +o 1c))"a) <-> E.y e. a ((x e. _V |-> (x +o 1c))` y) = z))
14	11, 12, 13	mp2an 653	. . . . . . . . . 10 |- (z e. ((x e. _V |-> (x +o 1c))"a) <-> E.y e. a ((x e. _V |-> (x +o 1c))` y) = z)
15	14	imbi1i 315	. . . . . . . . 9 |- ((z e. ((x e. _V |-> (x +o 1c))"a) -> z e. a) <-> (E.y e. a ((x e. _V |-> (x +o 1c))` y) = z -> z e. a))
16		r19.23v 2731	. . . . . . . . 9 |- (A.y e. a (((x e. _V |-> (x +o 1c))` y) = z -> z e. a) <-> (E.y e. a ((x e. _V |-> (x +o 1c))` y) = z -> z e. a))
17	15, 16	bitr4i 243	. . . . . . . 8 |- ((z e. ((x e. _V |-> (x +o 1c))"a) -> z e. a) <-> A.y e. a (((x e. _V |-> (x +o 1c))` y) = z -> z e. a))
18	17	albii 1566	. . . . . . 7 |- (A.z(z e. ((x e. _V |-> (x +o 1c))"a) -> z e. a) <-> A.zA.y e. a (((x e. _V |-> (x +o 1c))` y) = z -> z e. a))
19	4, 18	bitr4i 243	. . . . . 6 |- (A.y e. a A.z(((x e. _V |-> (x +o 1c))` y) = z -> z e. a) <-> A.z(z e. ((x e. _V |-> (x +o 1c))"a) -> z e. a))
20		vex 2863	. . . . . . . . . . . . 13 |- y e. _V
21		addceq1 4384	. . . . . . . . . . . . . 14 |- (x = y -> (x +o 1c) = (y +o 1c))
22	20, 8	addcex 4395	. . . . . . . . . . . . . 14 |- (y +o 1c) e. _V
23	21, 5, 22	fvmpt 5701	. . . . . . . . . . . . 13 |- (y e. _V -> ((x e. _V |-> (x +o 1c))` y) = (y +o 1c))
24	20, 23	ax-mp 5	. . . . . . . . . . . 12 |- ((x e. _V |-> (x +o 1c))` y) = (y +o 1c)
25	24	eqeq1i 2360	. . . . . . . . . . 11 |- (((x e. _V |-> (x +o 1c))` y) = z <-> (y +o 1c) = z)
26		eqcom 2355	. . . . . . . . . . 11 |- ((y +o 1c) = z <-> z = (y +o 1c))
27	25, 26	bitri 240	. . . . . . . . . 10 |- (((x e. _V |-> (x +o 1c))` y) = z <-> z = (y +o 1c))
28	27	imbi1i 315	. . . . . . . . 9 |- ((((x e. _V |-> (x +o 1c))` y) = z -> z e. a) <-> (z = (y +o 1c) -> z e. a))
29	28	albii 1566	. . . . . . . 8 |- (A.z(((x e. _V |-> (x +o 1c))` y) = z -> z e. a) <-> A.z(z = (y +o 1c) -> z e. a))
30		eleq1 2413	. . . . . . . . 9 |- (z = (y +o 1c) -> (z e. a <-> (y +o 1c) e. a))
31	22, 30	ceqsalv 2886	. . . . . . . 8 |- (A.z(z = (y +o 1c) -> z e. a) <-> (y +o 1c) e. a)
32	29, 31	bitri 240	. . . . . . 7 |- (A.z(((x e. _V |-> (x +o 1c))` y) = z -> z e. a) <-> (y +o 1c) e. a)
33	32	ralbii 2639	. . . . . 6 |- (A.y e. a A.z(((x e. _V |-> (x +o 1c))` y) = z -> z e. a) <-> A.y e. a (y +o 1c) e. a)
34	3, 19, 33	3bitr2ri 265	. . . . 5 |- (A.y e. a (y +o 1c) e. a <-> ((x e. _V |-> (x +o 1c))"a) C_ a)
35	2, 34	anbi12i 678	. . . 4 |- ((0c e. a /\ A.y e. a (y +o 1c) e. a) <-> ({0c} C_ a /\ ((x e. _V |-> (x +o 1c))"a) C_ a))
36	35	abbii 2466	. . 3 |- {a | (0c e. a /\ A.y e. a (y +o 1c) e. a)} = {a | ({0c} C_ a /\ ((x e. _V |-> (x +o 1c))"a) C_ a)}
37	36	inteqi 3931	. 2 |- |^|{a | (0c e. a /\ A.y e. a (y +o 1c) e. a)} = |^|{a | ({0c} C_ a /\ ((x e. _V |-> (x +o 1c))"a) C_ a)}
38		df-nnc 4380	. 2 |- Nn = |^|{a | (0c e. a /\ A.y e. a (y +o 1c) e. a)}
39		df-clos1 5874	. 2 |- Clos1 ({0c}, (x e. _V |-> (x +o 1c))) = |^|{a | ({0c} C_ a /\ ((x e. _V |-> (x +o 1c))"a) C_ a)}
40	37, 38, 39	3eqtr4i 2383	1 |- Nn = Clos1 ({0c}, (x e. _V |-> (x +o 1c)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  E.wrex 2616  _Vcvv 2860   C_ wss 3258  {csn 3738  |^|cint 3927  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o 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)