Theorem peano5nni 6071

Description: Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34.

Hypothesis

Ref	Expression
peano5nn.1	|- A e. V

Assertion

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

Distinct variable group:   x,A

Proof of Theorem peano5nni

Step	Hyp	Ref	Expression
1		peano5nn.1	. . . 4 |- A e. V
2		eleq2 1578	. . . . 5 |- (y = A -> (1 e. y <-> 1 e. A))
3		eleq2 1578	. . . . . 6 |- (y = A -> ((x + 1) e. y <-> (x + 1) e. A))
4	3	raleqd 1837	. . . . 5 |- (y = A -> (A.x e. y (x + 1) e. y <-> A.x e. A (x + 1) e. A))
5	2, 4	anbi12d 631	. . . 4 |- (y = A -> ((1 e. y /\ A.x e. y (x + 1) e. y) <-> (1 e. A /\ A.x e. A (x + 1) e. A)))
6	1, 5	elab 1943	. . 3 |- (A e. {y | (1 e. y /\ A.x e. y (x + 1) e. y)} <-> (1 e. A /\ A.x e. A (x + 1) e. A))
7		intss1 2615	. . 3 |- (A 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)} (_ A)
8	6, 7	sylbir 199	. 2 |- ((1 e. A /\ A.x e. A (x + 1) e. A) -> |^|{y | (1 e. y /\ A.x e. y (x + 1) e. y)} (_ A)
9		df-n 6070	. 2 |- NN = |^|{y | (1 e. y /\ A.x e. y (x + 1) e. y)}
10	8, 9	syl5ss 2157	1 |- ((1 e. A /\ A.x e. A (x + 1) e. A) -> NN (_ A)

Syntax hints:   -> wi 3   /\ wa 221   = wceq 992   e. wcel 994  {cab 1505  A.wral 1691  Vcvv 1857   (_ wss 2099  |^|cint 2600  (class class class)co 4021  1c1 5389   + caddc 5391  NNcn 5450

This theorem is referenced by:  nnssre 6072  nnind 6082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ral 1695  df-v 1858  df-in 2103  df-ss 2105  df-int 2601  df-n 6070

Copyright terms: Public domain