Theorem peano5nni 9205

Description: Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
peano5nni	|- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A )

Distinct variable group:   x, A

Proof of Theorem peano5nni

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 8238	. . . 4 |- 1 e. RR
2		elin 3392	. . . . 5 |- ( 1 e. ( A i^i RR ) <-> ( 1 e. A /\ 1 e. RR ) )
3	2	biimpri 133	. . . 4 |- ( ( 1 e. A /\ 1 e. RR ) -> 1 e. ( A i^i RR ) )
4	1, 3	mpan2 425	. . 3 |- ( 1 e. A -> 1 e. ( A i^i RR ) )
5		inss1 3429	. . . . 5 |- ( A i^i RR ) C_ A
6		ssralv 3292	. . . . 5 |- ( ( A i^i RR ) C_ A -> ( A. x e. A ( x + 1 ) e. A -> A. x e. ( A i^i RR ) ( x + 1 ) e. A ) )
7	5, 6	ax-mp 5	. . . 4 |- ( A. x e. A ( x + 1 ) e. A -> A. x e. ( A i^i RR ) ( x + 1 ) e. A )
8		inss2 3430	. . . . . . . 8 |- ( A i^i RR ) C_ RR
9	8	sseli 3224	. . . . . . 7 |- ( x e. ( A i^i RR ) -> x e. RR )
10		1red 8254	. . . . . . 7 |- ( x e. ( A i^i RR ) -> 1 e. RR )
11	9, 10	readdcld 8268	. . . . . 6 |- ( x e. ( A i^i RR ) -> ( x + 1 ) e. RR )
12		elin 3392	. . . . . . 7 |- ( ( x + 1 ) e. ( A i^i RR ) <-> ( ( x + 1 ) e. A /\ ( x + 1 ) e. RR ) )
13	12	simplbi2com 1490	. . . . . 6 |- ( ( x + 1 ) e. RR -> ( ( x + 1 ) e. A -> ( x + 1 ) e. ( A i^i RR ) ) )
14	11, 13	syl 14	. . . . 5 |- ( x e. ( A i^i RR ) -> ( ( x + 1 ) e. A -> ( x + 1 ) e. ( A i^i RR ) ) )
15	14	ralimia 2594	. . . 4 |- ( A. x e. ( A i^i RR ) ( x + 1 ) e. A -> A. x e. ( A i^i RR ) ( x + 1 ) e. ( A i^i RR ) )
16	7, 15	syl 14	. . 3 |- ( A. x e. A ( x + 1 ) e. A -> A. x e. ( A i^i RR ) ( x + 1 ) e. ( A i^i RR ) )
17		reex 8226	. . . . 5 |- RR e. _V
18	17	inex2 4229	. . . 4 |- ( A i^i RR ) e. _V
19		eleq2 2295	. . . . . . 7 |- ( y = ( A i^i RR ) -> ( 1 e. y <-> 1 e. ( A i^i RR ) ) )
20		eleq2 2295	. . . . . . . 8 |- ( y = ( A i^i RR ) -> ( ( x + 1 ) e. y <-> ( x + 1 ) e. ( A i^i RR ) ) )
21	20	raleqbi1dv 2743	. . . . . . 7 |- ( y = ( A i^i RR ) -> ( A. x e. y ( x + 1 ) e. y <-> A. x e. ( A i^i RR ) ( x + 1 ) e. ( A i^i RR ) ) )
22	19, 21	anbi12d 473	. . . . . 6 |- ( y = ( A i^i RR ) -> ( ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) <-> ( 1 e. ( A i^i RR ) /\ A. x e. ( A i^i RR ) ( x + 1 ) e. ( A i^i RR ) ) ) )
23	22	elabg 2953	. . . . 5 |- ( ( A i^i RR ) e. _V -> ( ( A i^i RR ) e. { y | ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) } <-> ( 1 e. ( A i^i RR ) /\ A. x e. ( A i^i RR ) ( x + 1 ) e. ( A i^i RR ) ) ) )
24		dfnn2 9204	. . . . . 6 |- NN = |^| { y | ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) }
25		intss1 3948	. . . . . 6 |- ( ( A i^i RR ) e. { y | ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) } -> |^| { y | ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) } C_ ( A i^i RR ) )
26	24, 25	eqsstrid 3274	. . . . 5 |- ( ( A i^i RR ) e. { y | ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) } -> NN C_ ( A i^i RR ) )
27	23, 26	biimtrrdi 164	. . . 4 |- ( ( A i^i RR ) e. _V -> ( ( 1 e. ( A i^i RR ) /\ A. x e. ( A i^i RR ) ( x + 1 ) e. ( A i^i RR ) ) -> NN C_ ( A i^i RR ) ) )
28	18, 27	ax-mp 5	. . 3 |- ( ( 1 e. ( A i^i RR ) /\ A. x e. ( A i^i RR ) ( x + 1 ) e. ( A i^i RR ) ) -> NN C_ ( A i^i RR ) )
29	4, 16, 28	syl2an 289	. 2 |- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ ( A i^i RR ) )
30	29, 5	sstrdi 3240	1 |- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   _Vcvv 2803   i^i cin 3200   C_ wss 3201   |^|cint 3933  (class class class)co 6028   RRcr 8091   1c1 8093   + caddc 8095   NNcn 9202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-int 3934  df-inn 9203

This theorem is referenced by:  nnssre  9206  nnind  9218