Theorem finds 4582

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 4114	. . . . . 6 |- (/) e. _V
3		finds.1	. . . . . 6 |- ( x = (/) -> ( ph <-> ps ) )
4	2, 3	elab 2874	. . . . 5 |- ( (/) e. { x | ph } <-> ps )
5	1, 4	mpbir 145	. . . 4 |- (/) e. { x | ph }
6		finds.6	. . . . . 6 |- ( y e. om -> ( ch -> th ) )
7		vex 2733	. . . . . . 7 |- y e. _V
8		finds.2	. . . . . . 7 |- ( x = y -> ( ph <-> ch ) )
9	7, 8	elab 2874	. . . . . 6 |- ( y e. { x | ph } <-> ch )
10	7	sucex 4481	. . . . . . 7 |- suc y e. _V
11		finds.3	. . . . . . 7 |- ( x = suc y -> ( ph <-> th ) )
12	10, 11	elab 2874	. . . . . 6 |- ( suc y e. { x | ph } <-> th )
13	6, 9, 12	3imtr4g 204	. . . . 5 |- ( y e. om -> ( y e. { x | ph } -> suc y e. { x | ph } ) )
14	13	rgen 2523	. . . 4 |- A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } )
15		peano5 4580	. . . 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 424	. . 3 |- om C_ { x | ph }
17	16	sseli 3143	. 2 |- ( A e. om -> A e. { x | ph } )
18		finds.4	. . 3 |- ( x = A -> ( ph <-> ta ) )
19	18	elabg 2876	. 2 |- ( A e. om -> ( A e. { x | ph } <-> ta ) )
20	17, 19	mpbid 146	1 |- ( A e. om -> ta )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   C_ wss 3121   (/)c0 3414   suc csuc 4348   omcom 4572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-int 3830  df-suc 4354  df-iom 4573

This theorem is referenced by:  findes  4585  nn0suc  4586  elomssom  4587  ordom  4589  nndceq0  4600  0elnn  4601  omsinds  4604  nna0r  6454  nnm0r  6455  nnsucelsuc  6467  nneneq  6831  php5  6832  php5dom  6837  fidcenumlemrk  6927  fidcenumlemr  6928  nnnninfeq  7100  nnnninfeq2  7101  frec2uzltd  10346  frecuzrdgg  10359  seq3val  10401  seqvalcd  10402  omgadd  10724  zfz1iso  10763  ennnfonelemhom  12357  nninfsellemdc  14003