Theorem tfinds3 3173

Description: Principle of Transfinite Induction (inference schema) with implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals.

Hypotheses

Ref	Expression
tfinds3.1	|- (x = (/) -> (ph <-> ps))
tfinds3.2	|- (x = y -> (ph <-> ch))
tfinds3.3	|- (x = suc y -> (ph <-> th))
tfinds3.4	|- (x = A -> (ph <-> ta))
tfinds3.5	|- (et -> ps)
tfinds3.6	|- (y e. On -> (et -> (ch -> th)))
tfinds3.7	|- (Lim x -> (et -> (A.y e. x ch -> ph)))

Assertion

Ref	Expression
tfinds3	|- (A e. On -> (et -> ta))

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

Proof of Theorem tfinds3

Step	Hyp	Ref	Expression
1		tfinds3.1	. . 3 |- (x = (/) -> (ph <-> ps))
2	1	imbi2d 614	. 2 |- (x = (/) -> ((et -> ph) <-> (et -> ps)))
3		tfinds3.2	. . 3 |- (x = y -> (ph <-> ch))
4	3	imbi2d 614	. 2 |- (x = y -> ((et -> ph) <-> (et -> ch)))
5		tfinds3.3	. . 3 |- (x = suc y -> (ph <-> th))
6	5	imbi2d 614	. 2 |- (x = suc y -> ((et -> ph) <-> (et -> th)))
7		tfinds3.4	. . 3 |- (x = A -> (ph <-> ta))
8	7	imbi2d 614	. 2 |- (x = A -> ((et -> ph) <-> (et -> ta)))
9		tfinds3.5	. 2 |- (et -> ps)
10		tfinds3.6	. . 3 |- (y e. On -> (et -> (ch -> th)))
11	10	a2d 13	. 2 |- (y e. On -> ((et -> ch) -> (et -> th)))
12		tfinds3.7	. . . 4 |- (Lim x -> (et -> (A.y e. x ch -> ph)))
13	12	a2d 13	. . 3 |- (Lim x -> ((et -> A.y e. x ch) -> (et -> ph)))
14		r19.21v 1719	. . 3 |- (A.y e. x (et -> ch) <-> (et -> A.y e. x ch))
15	13, 14	syl5ib 206	. 2 |- (Lim x -> (A.y e. x (et -> ch) -> (et -> ph)))
16	2, 4, 6, 8, 9, 11, 15	tfinds 3168	1 |- (A e. On -> (et -> ta))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  A.wral 1648  (/)c0 2284  Oncon0 2955  Lim wlim 2956  suc csuc 2957

This theorem is referenced by:  oacl 4177  omcl 4178  oecl 4179  oawordri 4191  oaass 4202  oarec 4203  omordi 4204  omwordri 4210  odi 4217  omass 4218  oen0 4220  oewordri 4226  oeworde 4227

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-if 2367  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-lim 2960  df-suc 2961

Copyright terms: Public domain