Theorem onfr 3702

Description: The ordinal class is founded. This lemma is needed for ordon 3863 in order to eliminate the need for the Axiom of Regularity.

Assertion

Ref	Expression
onfr	|- _E Fr On

Proof of Theorem onfr

Step	Hyp	Ref	Expression
1		dfepfr 3640	. 2 |- ( _E Fr On <-> A.x((x C_ On /\ x =/= (/)) -> E.z e. x (x i^i z) = (/)))
2		ineq2 2790	. . . . . . . . . 10 |- (z = y -> (x i^i z) = (x i^i y))
3	2	eqeq1d 1892	. . . . . . . . 9 |- (z = y -> ((x i^i z) = (/) <-> (x i^i y) = (/)))
4	3	rcla4ev 2381	. . . . . . . 8 |- ((y e. x /\ (x i^i y) = (/)) -> E.z e. x (x i^i z) = (/))
5	4	expcom 403	. . . . . . 7 |- ((x i^i y) = (/) -> (y e. x -> E.z e. x (x i^i z) = (/)))
6	5	a1d 15	. . . . . 6 |- ((x i^i y) = (/) -> (x C_ On -> (y e. x -> E.z e. x (x i^i z) = (/))))
7		ssel 2615	. . . . . . . . 9 |- (x C_ On -> (y e. x -> y e. On))
8		visset 2295	. . . . . . . . . 10 |- y e. _V
9	8	elon 3666	. . . . . . . . 9 |- (y e. On <-> Ord y)
10	7, 9	syl6ib 229	. . . . . . . 8 |- (x C_ On -> (y e. x -> Ord y))
11		ordfr 3673	. . . . . . . . 9 |- (Ord y -> _E Fr y)
12		ordtr 3672	. . . . . . . . 9 |- (Ord y -> Tr y)
13		inss2 2813	. . . . . . . . . . . 12 |- (x i^i y) C_ y
14		visset 2295	. . . . . . . . . . . . . 14 |- x e. _V
15	14	inex1 3452	. . . . . . . . . . . . 13 |- (x i^i y) e. _V
16	15	epfrc 3642	. . . . . . . . . . . 12 |- (( _E Fr y /\ (x i^i y) C_ y /\ (x i^i y) =/= (/)) -> E.z e. (x i^i y)((x i^i y) i^i z) = (/))
17	13, 16	mp3an2 1179	. . . . . . . . . . 11 |- (( _E Fr y /\ (x i^i y) =/= (/)) -> E.z e. (x i^i y)((x i^i y) i^i z) = (/))
18	17	ex 402	. . . . . . . . . 10 |- ( _E Fr y -> ((x i^i y) =/= (/) -> E.z e. (x i^i y)((x i^i y) i^i z) = (/)))
19		ax-17 1317	. . . . . . . . . . 11 |- (Tr y -> A.zTr y)
20		hbre1 2150	. . . . . . . . . . 11 |- (E.z e. x (x i^i z) = (/) -> A.zE.z e. x (x i^i z) = (/))
21		inss1 2812	. . . . . . . . . . . . . . . . . 18 |- (x i^i y) C_ x
22	21	sseli 2617	. . . . . . . . . . . . . . . . 17 |- (z e. (x i^i y) -> z e. x)
23		trss 3421	. . . . . . . . . . . . . . . . . . . 20 |- (Tr y -> (z e. y -> z C_ y))
24	13	sseli 2617	. . . . . . . . . . . . . . . . . . . 20 |- (z e. (x i^i y) -> z e. y)
25	23, 24	syl5 20	. . . . . . . . . . . . . . . . . . 19 |- (Tr y -> (z e. (x i^i y) -> z C_ y))
26		sseqin2 2811	. . . . . . . . . . . . . . . . . . . . . 22 |- (z C_ y <-> (y i^i z) = z)
27		ineq2 2790	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y i^i z) = z -> (x i^i (y i^i z)) = (x i^i z))
28		inass 2804	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((x i^i y) i^i z) = (x i^i (y i^i z))
29	27, 28	syl5req 1941	. . . . . . . . . . . . . . . . . . . . . 22 |- ((y i^i z) = z -> (x i^i z) = ((x i^i y) i^i z))
30	26, 29	sylbi 216	. . . . . . . . . . . . . . . . . . . . 21 |- (z C_ y -> (x i^i z) = ((x i^i y) i^i z))
31	30	eqeq1d 1892	. . . . . . . . . . . . . . . . . . . 20 |- (z C_ y -> ((x i^i z) = (/) <-> ((x i^i y) i^i z) = (/)))
32	31	biimprcd 173	. . . . . . . . . . . . . . . . . . 19 |- (((x i^i y) i^i z) = (/) -> (z C_ y -> (x i^i z) = (/)))
33	25, 32	sylan9 517	. . . . . . . . . . . . . . . . . 18 |- ((Tr y /\ ((x i^i y) i^i z) = (/)) -> (z e. (x i^i y) -> (x i^i z) = (/)))
34	33	imp 377	. . . . . . . . . . . . . . . . 17 |- (((Tr y /\ ((x i^i y) i^i z) = (/)) /\ z e. (x i^i y)) -> (x i^i z) = (/))
35	22, 34	anim12i 360	. . . . . . . . . . . . . . . 16 |- ((z e. (x i^i y) /\ ((Tr y /\ ((x i^i y) i^i z) = (/)) /\ z e. (x i^i y))) -> (z e. x /\ (x i^i z) = (/)))
36	35	exp32 408	. . . . . . . . . . . . . . 15 |- (z e. (x i^i y) -> ((Tr y /\ ((x i^i y) i^i z) = (/)) -> (z e. (x i^i y) -> (z e. x /\ (x i^i z) = (/)))))
37	36	pm2.43b 81	. . . . . . . . . . . . . 14 |- ((Tr y /\ ((x i^i y) i^i z) = (/)) -> (z e. (x i^i y) -> (z e. x /\ (x i^i z) = (/))))
38	37	ex 402	. . . . . . . . . . . . 13 |- (Tr y -> (((x i^i y) i^i z) = (/) -> (z e. (x i^i y) -> (z e. x /\ (x i^i z) = (/)))))
39	38	com23 36	. . . . . . . . . . . 12 |- (Tr y -> (z e. (x i^i y) -> (((x i^i y) i^i z) = (/) -> (z e. x /\ (x i^i z) = (/)))))
40		ra4e 2156	. . . . . . . . . . . 12 |- ((z e. x /\ (x i^i z) = (/)) -> E.z e. x (x i^i z) = (/))
41	39, 40	syl8 27	. . . . . . . . . . 11 |- (Tr y -> (z e. (x i^i y) -> (((x i^i y) i^i z) = (/) -> E.z e. x (x i^i z) = (/))))
42	19, 20, 41	r19.23ad 2213	. . . . . . . . . 10 |- (Tr y -> (E.z e. (x i^i y)((x i^i y) i^i z) = (/) -> E.z e. x (x i^i z) = (/)))
43	18, 42	sylan9 517	. . . . . . . . 9 |- (( _E Fr y /\ Tr y) -> ((x i^i y) =/= (/) -> E.z e. x (x i^i z) = (/)))
44	11, 12, 43	syl11anc 524	. . . . . . . 8 |- (Ord y -> ((x i^i y) =/= (/) -> E.z e. x (x i^i z) = (/)))
45	10, 44	syl6 25	. . . . . . 7 |- (x C_ On -> (y e. x -> ((x i^i y) =/= (/) -> E.z e. x (x i^i z) = (/))))
46	45	com3r 39	. . . . . 6 |- ((x i^i y) =/= (/) -> (x C_ On -> (y e. x -> E.z e. x (x i^i z) = (/))))
47	6, 46	pm2.61ine 2089	. . . . 5 |- (x C_ On -> (y e. x -> E.z e. x (x i^i z) = (/)))
48	47	19.23adv 1584	. . . 4 |- (x C_ On -> (E.y y e. x -> E.z e. x (x i^i z) = (/)))
49		n0 2884	. . . 4 |- (x =/= (/) <-> E.y y e. x)
50	48, 49	syl5ib 223	. . 3 |- (x C_ On -> (x =/= (/) -> E.z e. x (x i^i z) = (/)))
51	50	imp 377	. 2 |- ((x C_ On /\ x =/= (/)) -> E.z e. x (x i^i z) = (/))
52	1, 51	mpgbir 1334	1 |- _E Fr On

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  Tr wtr 3411   _E cep 3581   Fr wfr 3623  Ord word 3656  Oncon0 3657

This theorem is referenced by:  ordon 3863

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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