Theorem peano5nni 8881

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 7919	. . . 4 |- 1 e. RR
2		elin 3310	. . . . 5 |- ( 1 e. ( A i^i RR ) <-> ( 1 e. A /\ 1 e. RR ) )
3	2	biimpri 132	. . . 4 |- ( ( 1 e. A /\ 1 e. RR ) -> 1 e. ( A i^i RR ) )
4	1, 3	mpan2 423	. . 3 |- ( 1 e. A -> 1 e. ( A i^i RR ) )
5		inss1 3347	. . . . 5 |- ( A i^i RR ) C_ A
6		ssralv 3211	. . . . 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 3348	. . . . . . . 8 |- ( A i^i RR ) C_ RR
9	8	sseli 3143	. . . . . . 7 |- ( x e. ( A i^i RR ) -> x e. RR )
10		1red 7935	. . . . . . 7 |- ( x e. ( A i^i RR ) -> 1 e. RR )
11	9, 10	readdcld 7949	. . . . . 6 |- ( x e. ( A i^i RR ) -> ( x + 1 ) e. RR )
12		elin 3310	. . . . . . 7 |- ( ( x + 1 ) e. ( A i^i RR ) <-> ( ( x + 1 ) e. A /\ ( x + 1 ) e. RR ) )
13	12	simplbi2com 1437	. . . . . 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 2531	. . . 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 7908	. . . . 5 |- RR e. _V
18	17	inex2 4124	. . . 4 |- ( A i^i RR ) e. _V
19		eleq2 2234	. . . . . . 7 |- ( y = ( A i^i RR ) -> ( 1 e. y <-> 1 e. ( A i^i RR ) ) )
20		eleq2 2234	. . . . . . . 8 |- ( y = ( A i^i RR ) -> ( ( x + 1 ) e. y <-> ( x + 1 ) e. ( A i^i RR ) ) )
21	20	raleqbi1dv 2673	. . . . . . 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 470	. . . . . 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 2876	. . . . 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 8880	. . . . . 6 |- NN = |^| { y | ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) }
25		intss1 3846	. . . . . 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 3193	. . . . 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	syl6bir 163	. . . 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 287	. 2 |- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ ( A i^i RR ) )
30	29, 5	sstrdi 3159	1 |- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   _Vcvv 2730   i^i cin 3120   C_ wss 3121   |^|cint 3831  (class class class)co 5853   RRcr 7773   1c1 7775   + caddc 7777   NNcn 8878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-int 3832  df-inn 8879

This theorem is referenced by:  nnssre  8882  nnind  8894