Theorem peano5nn 5928

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
peano5nn	|- ((1 e. A /\ A.x e. A (x + 1) e. A) -> NN (_ A)

Distinct variable group:   x,A

Proof of Theorem peano5nn

Step	Hyp	Ref	Expression
1		peano5nn.1	. . . 4 |- A e. V
2		eleq2 1538	. . . . 5 |- (y = A -> (1 e. y <-> 1 e. A))
3		eleq2 1538	. . . . . 6 |- (y = A -> ((x + 1) e. y <-> (x + 1) e. A))
4	3	raleqd 1794	. . . . 5 |- (y = A -> (A.x e. y (x + 1) e. y <-> A.x e. A (x + 1) e. A))
5	2, 4	anbi12d 630	. . . 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 1900	. . 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 2552	. . 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 201	. 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 5927	. 2 |- NN = |^|{y | (1 e. y /\ A.x e. y (x + 1) e. y)}
10	8, 9	syl5ss 2108	1 |- ((1 e. A /\ A.x e. A (x + 1) e. A) -> NN (_ A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  A.wral 1648  Vcvv 1814   (_ wss 2050  |^|cint 2537  (class class class)co 3969  1c1 5247   + caddc 5249  NNcn 5308

This theorem is referenced by:  nnssre 5929  nnind 5939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-int 2538  df-n 5927

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