Theorem peano5 4410

Description: The principle of mathematical induction: a set containing cardinal zero and closed under the successor operator is a superset of the finite cardinals. Theorem X.1.6 of [Rosser] p. 276. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
peano5	|- ((A e. V /\ 0c e. A /\ A.x e. Nn (x e. A -> (x +o 1c) e. A)) -> Nn C_ A)

Distinct variable group:   x,A

Allowed substitution hint:   V(x)

Proof of Theorem peano5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nncex 4397	. . 3 |- Nn e. _V
2		inexg 4101	. . 3 |- (( Nn e. _V /\ A e. V) -> ( Nn i^i A) e. _V)
3	1, 2	mpan 651	. 2 |- (A e. V -> ( Nn i^i A) e. _V)
4		peano1 4403	. . 3 |- 0c e. Nn
5		elin 3220	. . . 4 |- (0c e. ( Nn i^i A) <-> (0c e. Nn /\ 0c e. A))
6	5	biimpri 197	. . 3 |- ((0c e. Nn /\ 0c e. A) -> 0c e. ( Nn i^i A))
7	4, 6	mpan 651	. 2 |- (0c e. A -> 0c e. ( Nn i^i A))
8		elin 3220	. . . . . 6 |- (x e. ( Nn i^i A) <-> (x e. Nn /\ x e. A))
9	8	imbi1i 315	. . . . 5 |- ((x e. ( Nn i^i A) -> (x +o 1c) e. A) <-> ((x e. Nn /\ x e. A) -> (x +o 1c) e. A))
10		impexp 433	. . . . 5 |- (((x e. Nn /\ x e. A) -> (x +o 1c) e. A) <-> (x e. Nn -> (x e. A -> (x +o 1c) e. A)))
11	9, 10	bitri 240	. . . 4 |- ((x e. ( Nn i^i A) -> (x +o 1c) e. A) <-> (x e. Nn -> (x e. A -> (x +o 1c) e. A)))
12		inss1 3476	. . . . . . . 8 |- ( Nn i^i A) C_ Nn
13	12	sseli 3270	. . . . . . 7 |- (x e. ( Nn i^i A) -> x e. Nn )
14		peano2 4404	. . . . . . 7 |- (x e. Nn -> (x +o 1c) e. Nn )
15	13, 14	syl 15	. . . . . 6 |- (x e. ( Nn i^i A) -> (x +o 1c) e. Nn )
16		elin 3220	. . . . . . . 8 |- ((x +o 1c) e. ( Nn i^i A) <-> ((x +o 1c) e. Nn /\ (x +o 1c) e. A))
17	16	biimpri 197	. . . . . . 7 |- (((x +o 1c) e. Nn /\ (x +o 1c) e. A) -> (x +o 1c) e. ( Nn i^i A))
18	17	a1i 10	. . . . . 6 |- (x e. ( Nn i^i A) -> (((x +o 1c) e. Nn /\ (x +o 1c) e. A) -> (x +o 1c) e. ( Nn i^i A)))
19	15, 18	mpand 656	. . . . 5 |- (x e. ( Nn i^i A) -> ((x +o 1c) e. A -> (x +o 1c) e. ( Nn i^i A)))
20	19	a2i 12	. . . 4 |- ((x e. ( Nn i^i A) -> (x +o 1c) e. A) -> (x e. ( Nn i^i A) -> (x +o 1c) e. ( Nn i^i A)))
21	11, 20	sylbir 204	. . 3 |- ((x e. Nn -> (x e. A -> (x +o 1c) e. A)) -> (x e. ( Nn i^i A) -> (x +o 1c) e. ( Nn i^i A)))
22	21	ralimi2 2687	. 2 |- (A.x e. Nn (x e. A -> (x +o 1c) e. A) -> A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A))
23		df-nnc 4380	. . . 4 |- Nn = |^|{y | (0c e. y /\ A.x e. y (x +o 1c) e. y)}
24		eleq2 2414	. . . . . . . . 9 |- (y = ( Nn i^i A) -> (0c e. y <-> 0c e. ( Nn i^i A)))
25		eleq2 2414	. . . . . . . . . 10 |- (y = ( Nn i^i A) -> ((x +o 1c) e. y <-> (x +o 1c) e. ( Nn i^i A)))
26	25	raleqbi1dv 2816	. . . . . . . . 9 |- (y = ( Nn i^i A) -> (A.x e. y (x +o 1c) e. y <-> A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A)))
27	24, 26	anbi12d 691	. . . . . . . 8 |- (y = ( Nn i^i A) -> ((0c e. y /\ A.x e. y (x +o 1c) e. y) <-> (0c e. ( Nn i^i A) /\ A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A))))
28	27	elabg 2987	. . . . . . 7 |- (( Nn i^i A) e. _V -> (( Nn i^i A) e. {y | (0c e. y /\ A.x e. y (x +o 1c) e. y)} <-> (0c e. ( Nn i^i A) /\ A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A))))
29	28	biimprd 214	. . . . . 6 |- (( Nn i^i A) e. _V -> ((0c e. ( Nn i^i A) /\ A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A)) -> ( Nn i^i A) e. {y | (0c e. y /\ A.x e. y (x +o 1c) e. y)}))
30	29	3impib 1149	. . . . 5 |- ((( Nn i^i A) e. _V /\ 0c e. ( Nn i^i A) /\ A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A)) -> ( Nn i^i A) e. {y | (0c e. y /\ A.x e. y (x +o 1c) e. y)})
31		intss1 3942	. . . . 5 |- (( Nn i^i A) e. {y | (0c e. y /\ A.x e. y (x +o 1c) e. y)} -> |^|{y | (0c e. y /\ A.x e. y (x +o 1c) e. y)} C_ ( Nn i^i A))
32	30, 31	syl 15	. . . 4 |- ((( Nn i^i A) e. _V /\ 0c e. ( Nn i^i A) /\ A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A)) -> |^|{y | (0c e. y /\ A.x e. y (x +o 1c) e. y)} C_ ( Nn i^i A))
33	23, 32	syl5eqss 3316	. . 3 |- ((( Nn i^i A) e. _V /\ 0c e. ( Nn i^i A) /\ A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A)) -> Nn C_ ( Nn i^i A))
34		inss2 3477	. . 3 |- ( Nn i^i A) C_ A
35	33, 34	syl6ss 3285	. 2 |- ((( Nn i^i A) e. _V /\ 0c e. ( Nn i^i A) /\ A.x e. ( Nn i^i A)(x +o 1c) e. ( Nn i^i A)) -> Nn C_ A)
36	3, 7, 22, 35	syl3an 1224	1 |- ((A e. V /\ 0c e. A /\ A.x e. Nn (x e. A -> (x +o 1c) e. A)) -> Nn C_ A)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  _Vcvv 2860   i^i cin 3209   C_ wss 3258  |^|cint 3927  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376

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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-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-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  findsd  4411  dmfrec  6317