Theorem peano5nn 4424

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

Hypothesis

Ref	Expression
peano5c.1	|- A e. V

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem peano5nn

Step	Hyp	Ref	Expression
1		peano5c.1	. . . 4 |- A e. V
2		eleq2 1150	. . . . 5 |- (y = A -> (1 e. y <-> 1 e. A))
3		eleq2 1150	. . . . . . 7 |- (y = A -> (x e. y <-> x e. A))
4		eleq2 1150	. . . . . . 7 |- (y = A -> ((x + 1) e. y <-> (x + 1) e. A))
5	3, 4	imbi12d 474	. . . . . 6 |- (y = A -> ((x e. y -> (x + 1) e. y) <-> (x e. A -> (x + 1) e. A)))
6	5	bialdv 935	. . . . 5 |- (y = A -> (A.x(x e. y -> (x + 1) e. y) <-> A.x(x e. A -> (x + 1) e. A)))
7	2, 6	anbi12d 476	. . . 4 |- (y = A -> ((1 e. y /\ A.x(x e. y -> (x + 1) e. y)) <-> (1 e. A /\ A.x(x e. A -> (x + 1) e. A))))
8	1, 7	elab 1415	. . 3 |- (A e. {y | (1 e. y /\ A.x(x e. y -> (x + 1) e. y))} <-> (1 e. A /\ A.x(x e. A -> (x + 1) e. A)))
9		intss1 1979	. . 3 |- (A e. {y | (1 e. y /\ A.x(x e. y -> (x + 1) e. y))} -> |^|{y | (1 e. y /\ A.x(x e. y -> (x + 1) e. y))} (_ A)
10	8, 9	sylbir 176	. 2 |- ((1 e. A /\ A.x(x e. A -> (x + 1) e. A)) -> |^|{y | (1 e. y /\ A.x(x e. y -> (x + 1) e. y))} (_ A)
11		df-n 4423	. 2 |- NN = |^|{y | (1 e. y /\ A.x(x e. y -> (x + 1) e. y))}
12	10, 11	syl5ss 1544	1 |- ((1 e. A /\ A.x(x e. A -> (x + 1) e. A)) -> NN (_ A)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  |^|cint 1965  (class class class)co 3001  1c1 4029   + caddc 4031  NNcn 4093

This theorem is referenced by:  nnssre 4425  nnind 4434

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-int 1966  df-n 4423

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