Theorem finds 4742

Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)

Hypotheses

Ref	Expression
finds.1	|- ( x = (/) -> ( ph <-> ps ) )
finds.2	|- ( x = y -> ( ph <-> ch ) )
finds.3	|- ( x = suc y -> ( ph <-> th ) )
finds.4	|- ( x = A -> ( ph <-> ta ) )
finds.5	|- ps
finds.6	|- ( y e. om -> ( ch -> th ) )

Assertion

Ref	Expression
finds	|- ( A e. om -> ta )

Distinct variable groups:   x, y   x, A   ps, x   ch, x   th, x   ta, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   th( y)   ta( y)   A( y)

Proof of Theorem finds

Step	Hyp	Ref	Expression
1		finds.5	. . . . 5 |- ps
2		0ex 4255	. . . . . 6 |- (/) e. _V
3		finds.1	. . . . . 6 |- ( x = (/) -> ( ph <-> ps ) )
4	2, 3	elab 2970	. . . . 5 |- ( (/) e. { x | ph } <-> ps )
5	1, 4	mpbir 146	. . . 4 |- (/) e. { x | ph }
6		finds.6	. . . . . 6 |- ( y e. om -> ( ch -> th ) )
7		vex 2824	. . . . . . 7 |- y e. _V
8		finds.2	. . . . . . 7 |- ( x = y -> ( ph <-> ch ) )
9	7, 8	elab 2970	. . . . . 6 |- ( y e. { x | ph } <-> ch )
10	7	sucex 4641	. . . . . . 7 |- suc y e. _V
11		finds.3	. . . . . . 7 |- ( x = suc y -> ( ph <-> th ) )
12	10, 11	elab 2970	. . . . . 6 |- ( suc y e. { x | ph } <-> th )
13	6, 9, 12	3imtr4g 205	. . . . 5 |- ( y e. om -> ( y e. { x | ph } -> suc y e. { x | ph } ) )
14	13	rgen 2603	. . . 4 |- A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } )
15		peano5 4740	. . . 4 |- ( ( (/) e. { x | ph } /\ A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } ) ) -> om C_ { x | ph } )
16	5, 14, 15	mp2an 430	. . 3 |- om C_ { x | ph }
17	16	sseli 3244	. 2 |- ( A e. om -> A e. { x | ph } )
18		finds.4	. . 3 |- ( x = A -> ( ph <-> ta ) )
19	18	elabg 2972	. 2 |- ( A e. om -> ( A e. { x | ph } <-> ta ) )
20	17, 19	mpbid 147	1 |- ( A e. om -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   C_ wss 3220   (/)c0 3520   suc csuc 4505   omcom 4732

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733

This theorem is referenced by:  findes  4745  nn0suc  4746  elomssom  4747  ordom  4749  nndceq0  4760  0elnn  4761  omsinds  4764  nna0r  6741  nnm0r  6742  nnsucelsuc  6754  nneneq  7148  php5  7149  php5dom  7154  fidcenumlemrk  7261  fidcenumlemr  7262  nninfninc  7453  nnnninfeq  7458  nnnninfeq2  7459  frec2uzltd  10818  frecuzrdgg  10831  seq3val  10875  seqvalcd  10876  omgadd  11220  zfz1iso  11271  ennnfonelemhom  13284  nninfsellemdc  16958  nnnninfex  16970