Theorem onfr 5763

Description: The ordinal class is well-founded. This lemma is needed for ordon 6982 in order to eliminate the need for the Axiom of Regularity. (Contributed by NM, 17-May-1994.)

Assertion

Ref	Expression
onfr	|- _E Fr On

Proof of Theorem onfr

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfepfr 5099	. 2 |- ( _E Fr On <-> A. x ( ( x C_ On /\ x =/= (/) ) -> E. z e. x ( x i^i z ) = (/) ) )
2		n0 3931	. . . 4 |- ( x =/= (/) <-> E. y y e. x )
3		ineq2 3808	. . . . . . . . . 10 |- ( z = y -> ( x i^i z ) = ( x i^i y ) )
4	3	eqeq1d 2624	. . . . . . . . 9 |- ( z = y -> ( ( x i^i z ) = (/) <-> ( x i^i y ) = (/) ) )
5	4	rspcev 3309	. . . . . . . 8 |- ( ( y e. x /\ ( x i^i y ) = (/) ) -> E. z e. x ( x i^i z ) = (/) )
6	5	adantll 750	. . . . . . 7 |- ( ( ( x C_ On /\ y e. x ) /\ ( x i^i y ) = (/) ) -> E. z e. x ( x i^i z ) = (/) )
7		inss1 3833	. . . . . . . 8 |- ( x i^i y ) C_ x
8		ssel2 3598	. . . . . . . . . . 11 |- ( ( x C_ On /\ y e. x ) -> y e. On )
9		eloni 5733	. . . . . . . . . . 11 |- ( y e. On -> Ord y )
10		ordfr 5738	. . . . . . . . . . 11 |- ( Ord y -> _E Fr y )
11	8, 9, 10	3syl 18	. . . . . . . . . 10 |- ( ( x C_ On /\ y e. x ) -> _E Fr y )
12		inss2 3834	. . . . . . . . . . 11 |- ( x i^i y ) C_ y
13		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
14	13	inex1 4799	. . . . . . . . . . . 12 |- ( x i^i y ) e. _V
15	14	epfrc 5100	. . . . . . . . . . 11 |- ( ( _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 ) = (/) )
16	12, 15	mp3an2 1412	. . . . . . . . . 10 |- ( ( _E Fr y /\ ( x i^i y ) =/= (/) ) -> E. z e. ( x i^i y ) ( ( x i^i y ) i^i z ) = (/) )
17	11, 16	sylan 488	. . . . . . . . 9 |- ( ( ( x C_ On /\ y e. x ) /\ ( x i^i y ) =/= (/) ) -> E. z e. ( x i^i y ) ( ( x i^i y ) i^i z ) = (/) )
18		inass 3823	. . . . . . . . . . . . 13 |- ( ( x i^i y ) i^i z ) = ( x i^i ( y i^i z ) )
19	8, 9	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( x C_ On /\ y e. x ) -> Ord y )
20		elinel2 3800	. . . . . . . . . . . . . . . 16 |- ( z e. ( x i^i y ) -> z e. y )
21		ordelss 5739	. . . . . . . . . . . . . . . 16 |- ( ( Ord y /\ z e. y ) -> z C_ y )
22	19, 20, 21	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( ( x C_ On /\ y e. x ) /\ z e. ( x i^i y ) ) -> z C_ y )
23		sseqin2 3817	. . . . . . . . . . . . . . 15 |- ( z C_ y <-> ( y i^i z ) = z )
24	22, 23	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( x C_ On /\ y e. x ) /\ z e. ( x i^i y ) ) -> ( y i^i z ) = z )
25	24	ineq2d 3814	. . . . . . . . . . . . 13 |- ( ( ( x C_ On /\ y e. x ) /\ z e. ( x i^i y ) ) -> ( x i^i ( y i^i z ) ) = ( x i^i z ) )
26	18, 25	syl5eq 2668	. . . . . . . . . . . 12 |- ( ( ( x C_ On /\ y e. x ) /\ z e. ( x i^i y ) ) -> ( ( x i^i y ) i^i z ) = ( x i^i z ) )
27	26	eqeq1d 2624	. . . . . . . . . . 11 |- ( ( ( x C_ On /\ y e. x ) /\ z e. ( x i^i y ) ) -> ( ( ( x i^i y ) i^i z ) = (/) <-> ( x i^i z ) = (/) ) )
28	27	rexbidva 3049	. . . . . . . . . 10 |- ( ( x C_ On /\ y e. x ) -> ( E. z e. ( x i^i y ) ( ( x i^i y ) i^i z ) = (/) <-> E. z e. ( x i^i y ) ( x i^i z ) = (/) ) )
29	28	adantr 481	. . . . . . . . 9 |- ( ( ( x C_ On /\ y e. x ) /\ ( x i^i y ) =/= (/) ) -> ( E. z e. ( x i^i y ) ( ( x i^i y ) i^i z ) = (/) <-> E. z e. ( x i^i y ) ( x i^i z ) = (/) ) )
30	17, 29	mpbid 222	. . . . . . . 8 |- ( ( ( x C_ On /\ y e. x ) /\ ( x i^i y ) =/= (/) ) -> E. z e. ( x i^i y ) ( x i^i z ) = (/) )
31		ssrexv 3667	. . . . . . . 8 |- ( ( x i^i y ) C_ x -> ( E. z e. ( x i^i y ) ( x i^i z ) = (/) -> E. z e. x ( x i^i z ) = (/) ) )
32	7, 30, 31	mpsyl 68	. . . . . . 7 |- ( ( ( x C_ On /\ y e. x ) /\ ( x i^i y ) =/= (/) ) -> E. z e. x ( x i^i z ) = (/) )
33	6, 32	pm2.61dane 2881	. . . . . 6 |- ( ( x C_ On /\ y e. x ) -> E. z e. x ( x i^i z ) = (/) )
34	33	ex 450	. . . . 5 |- ( x C_ On -> ( y e. x -> E. z e. x ( x i^i z ) = (/) ) )
35	34	exlimdv 1861	. . . 4 |- ( x C_ On -> ( E. y y e. x -> E. z e. x ( x i^i z ) = (/) ) )
36	2, 35	syl5bi 232	. . 3 |- ( x C_ On -> ( x =/= (/) -> E. z e. x ( x i^i z ) = (/) ) )
37	36	imp 445	. 2 |- ( ( x C_ On /\ x =/= (/) ) -> E. z e. x ( x i^i z ) = (/) )
38	1, 37	mpgbir 1726	1 |- _E Fr On

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   _E cep 5028   Fr wfr 5070   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  ordon  6982