Theorem onfr 2237

Description: The ordinal class is founded. This lemma is needed for ordon 2238 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 2184	. 2 |- (E Fr On <-> A.x((x (_ On /\ -. x = (/)) -> E.z e. x (x i^i z) = (/)))
2		ineq2 1639	. . . . . . . . . . 11 |- (z = y -> (x i^i z) = (x i^i y))
3	2	cleq1d 1109	. . . . . . . . . 10 |- (z = y -> ((x i^i z) = (/) <-> (x i^i y) = (/)))
4	3	rcla4ev 1403	. . . . . . . . 9 |- ((y e. x /\ (x i^i y) = (/)) -> E.z e. x (x i^i z) = (/))
5	4	exp 291	. . . . . . . 8 |- (y e. x -> ((x i^i y) = (/) -> E.z e. x (x i^i z) = (/)))
6	5	com12 13	. . . . . . 7 |- ((x i^i y) = (/) -> (y e. x -> E.z e. x (x i^i z) = (/)))
7	6	a1d 14	. . . . . 6 |- ((x i^i y) = (/) -> (x (_ On -> (y e. x -> E.z e. x (x i^i z) = (/))))
8		ssel 1502	. . . . . . . . 9 |- (x (_ On -> (y e. x -> y e. On))
9		visset 1350	. . . . . . . . . 10 |- y e. V
10	9	elon 2208	. . . . . . . . 9 |- (y e. On <-> Ord y)
11	8, 10	syl6ib 185	. . . . . . . 8 |- (x (_ On -> (y e. x -> Ord y))
12		inss2 1658	. . . . . . . . . . 11 |- (x i^i y) (_ y
13		visset 1350	. . . . . . . . . . . . . 14 |- x e. V
14	13	inex1 1697	. . . . . . . . . . . . 13 |- (x i^i y) e. V
15	14	epfrc 2185	. . . . . . . . . . . 12 |- ((E Fr y /\ ((x i^i y) (_ y /\ -. (x i^i y) = (/))) -> E.z e. (x i^i y)((x i^i y) i^i z) = (/))
16	15	exp 291	. . . . . . . . . . 11 |- (E Fr y -> (((x i^i y) (_ y /\ -. (x i^i y) = (/)) -> E.z e. (x i^i y)((x i^i y) i^i z) = (/)))
17	12, 16	mpani 521	. . . . . . . . . 10 |- (E Fr y -> (-. (x i^i y) = (/) -> E.z e. (x i^i y)((x i^i y) i^i z) = (/)))
18		ax-17 925	. . . . . . . . . . 11 |- (Tr y -> A.zTr y)
19		hbre1 1239	. . . . . . . . . . 11 |- (E.z e. x (x i^i z) = (/) -> A.zE.z e. x (x i^i z) = (/))
20		inss1 1657	. . . . . . . . . . . . . . . . . 18 |- (x i^i y) (_ x
21	20	sseli 1504	. . . . . . . . . . . . . . . . 17 |- (z e. (x i^i y) -> z e. x)
22		trss 2050	. . . . . . . . . . . . . . . . . . . 20 |- (Tr y -> (z e. y -> z (_ y))
23	12	sseli 1504	. . . . . . . . . . . . . . . . . . . 20 |- (z e. (x i^i y) -> z e. y)
24	22, 23	syl5 22	. . . . . . . . . . . . . . . . . . 19 |- (Tr y -> (z e. (x i^i y) -> z (_ y))
25		sseqin2 1656	. . . . . . . . . . . . . . . . . . . . . 22 |- (z (_ y <-> (y i^i z) = z)
26		ineq2 1639	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y i^i z) = z -> (x i^i (y i^i z)) = (x i^i z))
27		inass 1650	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((x i^i y) i^i z) = (x i^i (y i^i z))
28	26, 27	syl5req 1137	. . . . . . . . . . . . . . . . . . . . . 22 |- ((y i^i z) = z -> (x i^i z) = ((x i^i y) i^i z))
29	25, 28	sylbi 174	. . . . . . . . . . . . . . . . . . . . 21 |- (z (_ y -> (x i^i z) = ((x i^i y) i^i z))
30	29	cleq1d 1109	. . . . . . . . . . . . . . . . . . . 20 |- (z (_ y -> ((x i^i z) = (/) <-> ((x i^i y) i^i z) = (/)))
31	30	biimprcd 138	. . . . . . . . . . . . . . . . . . 19 |- (((x i^i y) i^i z) = (/) -> (z (_ y -> (x i^i z) = (/)))
32	24, 31	sylan9 359	. . . . . . . . . . . . . . . . . 18 |- ((Tr y /\ ((x i^i y) i^i z) = (/)) -> (z e. (x i^i y) -> (x i^i z) = (/)))
33	32	imp 277	. . . . . . . . . . . . . . . . 17 |- (((Tr y /\ ((x i^i y) i^i z) = (/)) /\ z e. (x i^i y)) -> (x i^i z) = (/))
34	21, 33	anim12i 268	. . . . . . . . . . . . . . . 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) = (/)))
35	34	exp32 294	. . . . . . . . . . . . . . 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) = (/)))))
36	35	pm2.43b 61	. . . . . . . . . . . . . 14 |- ((Tr y /\ ((x i^i y) i^i z) = (/)) -> (z e. (x i^i y) -> (z e. x /\ (x i^i z) = (/))))
37	36	exp 291	. . . . . . . . . . . . 13 |- (Tr y -> (((x i^i y) i^i z) = (/) -> (z e. (x i^i y) -> (z e. x /\ (x i^i z) = (/)))))
38	37	com23 32	. . . . . . . . . . . 12 |- (Tr y -> (z e. (x i^i y) -> (((x i^i y) i^i z) = (/) -> (z e. x /\ (x i^i z) = (/)))))
39		ra4e 1244	. . . . . . . . . . . 12 |- ((z e. x /\ (x i^i z) = (/)) -> E.z e. x (x i^i z) = (/))
40	38, 39	syl8 25	. . . . . . . . . . 11 |- (Tr y -> (z e. (x i^i y) -> (((x i^i y) i^i z) = (/) -> E.z e. x (x i^i z) = (/))))
41	18, 19, 40	r19.23ad 1285	. . . . . . . . . 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) = (/)))
42	17, 41	sylan9 359	. . . . . . . . 9 |- ((E Fr y /\ Tr y) -> (-. (x i^i y) = (/) -> E.z e. x (x i^i z) = (/)))
43		ordfr 2214	. . . . . . . . 9 |- (Ord y -> E Fr y)
44		ordtr 2213	. . . . . . . . 9 |- (Ord y -> Tr y)
45	42, 43, 44	sylanc 361	. . . . . . . 8 |- (Ord y -> (-. (x i^i y) = (/) -> E.z e. x (x i^i z) = (/)))
46	11, 45	syl6 23	. . . . . . 7 |- (x (_ On -> (y e. x -> (-. (x i^i y) = (/) -> E.z e. x (x i^i z) = (/))))
47	46	com3r 35	. . . . . 6 |- (-. (x i^i y) = (/) -> (x (_ On -> (y e. x -> E.z e. x (x i^i z) = (/))))
48	7, 47	pm2.61i 110	. . . . 5 |- (x (_ On -> (y e. x -> E.z e. x (x i^i z) = (/)))
49	48	19.23adv 954	. . . 4 |- (x (_ On -> (E.y y e. x -> E.z e. x (x i^i z) = (/)))
50		n0 1714	. . . 4 |- (-. x = (/) <-> E.y y e. x)
51	49, 50	syl5ib 181	. . 3 |- (x (_ On -> (-. x = (/) -> E.z e. x (x i^i z) = (/)))
52	51	imp 277	. 2 |- ((x (_ On /\ -. x = (/)) -> E.z e. x (x i^i z) = (/))
53	1, 52	mpgbir 686	1 |- E Fr On

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  E.wex 678   = weq 797   e. wel 803   = wceq 1091   e. wcel 1092  E.wrex 1202   i^i cin 1486   (_ wss 1487  (/)c0 1707  Tr wtr 2041  Ecep 2056   Fr wfr 2061  Ord word 2198  Oncon0 2199

This theorem is referenced by:  ordon 2238

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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