Theorem finds 4727

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 4242	. . . . . 6 |- (/) e. _V
3		finds.1	. . . . . 6 |- ( x = (/) -> ( ph <-> ps ) )
4	2, 3	elab 2964	. . . . 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 2818	. . . . . . 7 |- y e. _V
8		finds.2	. . . . . . 7 |- ( x = y -> ( ph <-> ch ) )
9	7, 8	elab 2964	. . . . . 6 |- ( y e. { x | ph } <-> ch )
10	7	sucex 4626	. . . . . . 7 |- suc y e. _V
11		finds.3	. . . . . . 7 |- ( x = suc y -> ( ph <-> th ) )
12	10, 11	elab 2964	. . . . . 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 2597	. . . 4 |- A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } )
15		peano5 4725	. . . 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 426	. . 3 |- om C_ { x | ph }
17	16	sseli 3238	. 2 |- ( A e. om -> A e. { x | ph } )
18		finds.4	. . 3 |- ( x = A -> ( ph <-> ta ) )
19	18	elabg 2966	. 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 1398   e. wcel 2205   {cab 2220   A.wral 2522   C_ wss 3214   (/)c0 3512   suc csuc 4491   omcom 4717

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-suc 4497  df-iom 4718

This theorem is referenced by:  findes  4730  nn0suc  4731  elomssom  4732  ordom  4734  nndceq0  4745  0elnn  4746  omsinds  4749  nna0r  6724  nnm0r  6725  nnsucelsuc  6737  nneneq  7124  php5  7125  php5dom  7130  fidcenumlemrk  7237  fidcenumlemr  7238  nninfninc  7427  nnnninfeq  7432  nnnninfeq2  7433  frec2uzltd  10789  frecuzrdgg  10802  seq3val  10846  seqvalcd  10847  omgadd  11191  zfz1iso  11238  ennnfonelemhom  13250  nninfsellemdc  16914  nnnninfex  16926