Theorem peano5nni 8987

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 8020	. . . 4 |- 1 e. RR
2		elin 3343	. . . . 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 3380	. . . . 5 |- ( A i^i RR ) C_ A
6		ssralv 3244	. . . . 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 3381	. . . . . . . 8 |- ( A i^i RR ) C_ RR
9	8	sseli 3176	. . . . . . 7 |- ( x e. ( A i^i RR ) -> x e. RR )
10		1red 8036	. . . . . . 7 |- ( x e. ( A i^i RR ) -> 1 e. RR )
11	9, 10	readdcld 8051	. . . . . 6 |- ( x e. ( A i^i RR ) -> ( x + 1 ) e. RR )
12		elin 3343	. . . . . . 7 |- ( ( x + 1 ) e. ( A i^i RR ) <-> ( ( x + 1 ) e. A /\ ( x + 1 ) e. RR ) )
13	12	simplbi2com 1455	. . . . . 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 2555	. . . 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 8008	. . . . 5 |- RR e. _V
18	17	inex2 4165	. . . 4 |- ( A i^i RR ) e. _V
19		eleq2 2257	. . . . . . 7 |- ( y = ( A i^i RR ) -> ( 1 e. y <-> 1 e. ( A i^i RR ) ) )
20		eleq2 2257	. . . . . . . 8 |- ( y = ( A i^i RR ) -> ( ( x + 1 ) e. y <-> ( x + 1 ) e. ( A i^i RR ) ) )
21	20	raleqbi1dv 2702	. . . . . . 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 2907	. . . . 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 8986	. . . . . 6 |- NN = |^| { y | ( 1 e. y /\ A. x e. y ( x + 1 ) e. y ) }
25		intss1 3886	. . . . . 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 3226	. . . . 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 3192	1 |- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   _Vcvv 2760   i^i cin 3153   C_ wss 3154   |^|cint 3871  (class class class)co 5919   RRcr 7873   1c1 7875   + caddc 7877   NNcn 8984

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4148  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3160  df-ss 3167  df-int 3872  df-inn 8985

This theorem is referenced by:  nnssre  8988  nnind  9000