Theorem finds 4666

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 4187	. . . . . 6 |- (/) e. _V
3		finds.1	. . . . . 6 |- ( x = (/) -> ( ph <-> ps ) )
4	2, 3	elab 2924	. . . . 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 2779	. . . . . . 7 |- y e. _V
8		finds.2	. . . . . . 7 |- ( x = y -> ( ph <-> ch ) )
9	7, 8	elab 2924	. . . . . 6 |- ( y e. { x | ph } <-> ch )
10	7	sucex 4565	. . . . . . 7 |- suc y e. _V
11		finds.3	. . . . . . 7 |- ( x = suc y -> ( ph <-> th ) )
12	10, 11	elab 2924	. . . . . 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 2561	. . . 4 |- A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } )
15		peano5 4664	. . . 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 3197	. 2 |- ( A e. om -> A e. { x | ph } )
18		finds.4	. . 3 |- ( x = A -> ( ph <-> ta ) )
19	18	elabg 2926	. 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 1373   e. wcel 2178   {cab 2193   A.wral 2486   C_ wss 3174   (/)c0 3468   suc csuc 4430   omcom 4656

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-suc 4436  df-iom 4657

This theorem is referenced by:  findes  4669  nn0suc  4670  elomssom  4671  ordom  4673  nndceq0  4684  0elnn  4685  omsinds  4688  nna0r  6587  nnm0r  6588  nnsucelsuc  6600  nneneq  6979  php5  6980  php5dom  6985  fidcenumlemrk  7082  fidcenumlemr  7083  nninfninc  7251  nnnninfeq  7256  nnnninfeq2  7257  frec2uzltd  10585  frecuzrdgg  10598  seq3val  10642  seqvalcd  10643  omgadd  10984  zfz1iso  11023  ennnfonelemhom  12901  nninfsellemdc  16149  nnnninfex  16161