Theorem tfinds3 3166

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 612	. 2 |- (x = (/) -> ((et -> ph) <-> (et -> ps)))
3		tfinds3.2	. . 3 |- (x = y -> (ph <-> ch))
4	3	imbi2d 612	. 2 |- (x = y -> ((et -> ph) <-> (et -> ch)))
5		tfinds3.3	. . 3 |- (x = suc y -> (ph <-> th))
6	5	imbi2d 612	. 2 |- (x = suc y -> ((et -> ph) <-> (et -> th)))
7		tfinds3.4	. . 3 |- (x = A -> (ph <-> ta))
8	7	imbi2d 612	. 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 1716	. . 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 3161	1 |- (A e. On -> (et -> ta))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645  (/)c0 2280  Oncon0 2948  Lim wlim 2949  suc csuc 2950

This theorem is referenced by:  oacl 4170  omcl 4171  oecl 4172  oawordri 4184  oaass 4195  oarec 4196  omordi 4197  omwordri 4203  odi 4210  omass 4211  oen0 4213  oewordri 4219  oeworde 4220

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954

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