Theorem oawordeulem 7634

Description: Lemma for oawordex 7637. (Contributed by NM, 11-Dec-2004.)

Hypotheses

Ref	Expression
oawordeulem.1	|- A e. On
oawordeulem.2	|- B e. On
oawordeulem.3	|- S = { y e. On | B C_ ( A +o y ) }

Assertion

Ref	Expression
oawordeulem	|- ( A C_ B -> E! x e. On ( A +o x ) = B )

Distinct variable groups:   x, y, A   x, B, y   x, S

Allowed substitution hint:   S( y)

Proof of Theorem oawordeulem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oawordeulem.3	. . . . . 6 |- S = { y e. On | B C_ ( A +o y ) }
2		ssrab2 3687	. . . . . 6 |- { y e. On | B C_ ( A +o y ) } C_ On
3	1, 2	eqsstri 3635	. . . . 5 |- S C_ On
4		oawordeulem.2	. . . . . . 7 |- B e. On
5		oawordeulem.1	. . . . . . . 8 |- A e. On
6		oaword2 7633	. . . . . . . 8 |- ( ( B e. On /\ A e. On ) -> B C_ ( A +o B ) )
7	4, 5, 6	mp2an 708	. . . . . . 7 |- B C_ ( A +o B )
8		oveq2 6658	. . . . . . . . 9 |- ( y = B -> ( A +o y ) = ( A +o B ) )
9	8	sseq2d 3633	. . . . . . . 8 |- ( y = B -> ( B C_ ( A +o y ) <-> B C_ ( A +o B ) ) )
10	9, 1	elrab2 3366	. . . . . . 7 |- ( B e. S <-> ( B e. On /\ B C_ ( A +o B ) ) )
11	4, 7, 10	mpbir2an 955	. . . . . 6 |- B e. S
12	11	ne0ii 3923	. . . . 5 |- S =/= (/)
13		oninton 7000	. . . . 5 |- ( ( S C_ On /\ S =/= (/) ) -> |^| S e. On )
14	3, 12, 13	mp2an 708	. . . 4 |- |^| S e. On
15		onzsl 7046	. . . . . . . 8 |- ( |^| S e. On <-> ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) )
16	14, 15	mpbi 220	. . . . . . 7 |- ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) )
17		oveq2 6658	. . . . . . . . . . 11 |- ( |^| S = (/) -> ( A +o |^| S ) = ( A +o (/) ) )
18		oa0 7596	. . . . . . . . . . . 12 |- ( A e. On -> ( A +o (/) ) = A )
19	5, 18	ax-mp 5	. . . . . . . . . . 11 |- ( A +o (/) ) = A
20	17, 19	syl6eq 2672	. . . . . . . . . 10 |- ( |^| S = (/) -> ( A +o |^| S ) = A )
21	20	sseq1d 3632	. . . . . . . . 9 |- ( |^| S = (/) -> ( ( A +o |^| S ) C_ B <-> A C_ B ) )
22	21	biimprd 238	. . . . . . . 8 |- ( |^| S = (/) -> ( A C_ B -> ( A +o |^| S ) C_ B ) )
23		oveq2 6658	. . . . . . . . . . . 12 |- ( |^| S = suc z -> ( A +o |^| S ) = ( A +o suc z ) )
24		oasuc 7604	. . . . . . . . . . . . 13 |- ( ( A e. On /\ z e. On ) -> ( A +o suc z ) = suc ( A +o z ) )
25	5, 24	mpan 706	. . . . . . . . . . . 12 |- ( z e. On -> ( A +o suc z ) = suc ( A +o z ) )
26	23, 25	sylan9eqr 2678	. . . . . . . . . . 11 |- ( ( z e. On /\ |^| S = suc z ) -> ( A +o |^| S ) = suc ( A +o z ) )
27		vex 3203	. . . . . . . . . . . . . . 15 |- z e. _V
28	27	sucid 5804	. . . . . . . . . . . . . 14 |- z e. suc z
29		eleq2 2690	. . . . . . . . . . . . . 14 |- ( |^| S = suc z -> ( z e. |^| S <-> z e. suc z ) )
30	28, 29	mpbiri 248	. . . . . . . . . . . . 13 |- ( |^| S = suc z -> z e. |^| S )
31	14	oneli 5835	. . . . . . . . . . . . . 14 |- ( z e. |^| S -> z e. On )
32	1	inteqi 4479	. . . . . . . . . . . . . . . . 17 |- |^| S = |^| { y e. On | B C_ ( A +o y ) }
33	32	eleq2i 2693	. . . . . . . . . . . . . . . 16 |- ( z e. |^| S <-> z e. |^| { y e. On | B C_ ( A +o y ) } )
34		oveq2 6658	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( A +o y ) = ( A +o z ) )
35	34	sseq2d 3633	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( B C_ ( A +o y ) <-> B C_ ( A +o z ) ) )
36	35	onnminsb 7004	. . . . . . . . . . . . . . . 16 |- ( z e. On -> ( z e. |^| { y e. On | B C_ ( A +o y ) } -> -. B C_ ( A +o z ) ) )
37	33, 36	syl5bi 232	. . . . . . . . . . . . . . 15 |- ( z e. On -> ( z e. |^| S -> -. B C_ ( A +o z ) ) )
38		oacl 7615	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. On /\ z e. On ) -> ( A +o z ) e. On )
39	5, 38	mpan 706	. . . . . . . . . . . . . . . . 17 |- ( z e. On -> ( A +o z ) e. On )
40		ontri1 5757	. . . . . . . . . . . . . . . . 17 |- ( ( B e. On /\ ( A +o z ) e. On ) -> ( B C_ ( A +o z ) <-> -. ( A +o z ) e. B ) )
41	4, 39, 40	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( z e. On -> ( B C_ ( A +o z ) <-> -. ( A +o z ) e. B ) )
42	41	con2bid 344	. . . . . . . . . . . . . . 15 |- ( z e. On -> ( ( A +o z ) e. B <-> -. B C_ ( A +o z ) ) )
43	37, 42	sylibrd 249	. . . . . . . . . . . . . 14 |- ( z e. On -> ( z e. |^| S -> ( A +o z ) e. B ) )
44	31, 43	mpcom 38	. . . . . . . . . . . . 13 |- ( z e. |^| S -> ( A +o z ) e. B )
45	4	onordi 5832	. . . . . . . . . . . . . 14 |- Ord B
46		ordsucss 7018	. . . . . . . . . . . . . 14 |- ( Ord B -> ( ( A +o z ) e. B -> suc ( A +o z ) C_ B ) )
47	45, 46	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( A +o z ) e. B -> suc ( A +o z ) C_ B )
48	30, 44, 47	3syl 18	. . . . . . . . . . . 12 |- ( |^| S = suc z -> suc ( A +o z ) C_ B )
49	48	adantl 482	. . . . . . . . . . 11 |- ( ( z e. On /\ |^| S = suc z ) -> suc ( A +o z ) C_ B )
50	26, 49	eqsstrd 3639	. . . . . . . . . 10 |- ( ( z e. On /\ |^| S = suc z ) -> ( A +o |^| S ) C_ B )
51	50	rexlimiva 3028	. . . . . . . . 9 |- ( E. z e. On |^| S = suc z -> ( A +o |^| S ) C_ B )
52	51	a1d 25	. . . . . . . 8 |- ( E. z e. On |^| S = suc z -> ( A C_ B -> ( A +o |^| S ) C_ B ) )
53		oalim 7612	. . . . . . . . . . 11 |- ( ( A e. On /\ ( |^| S e. _V /\ Lim |^| S ) ) -> ( A +o |^| S ) = U_ z e. |^| S ( A +o z ) )
54	5, 53	mpan 706	. . . . . . . . . 10 |- ( ( |^| S e. _V /\ Lim |^| S ) -> ( A +o |^| S ) = U_ z e. |^| S ( A +o z ) )
55		iunss 4561	. . . . . . . . . . 11 |- ( U_ z e. |^| S ( A +o z ) C_ B <-> A. z e. |^| S ( A +o z ) C_ B )
56	4	onelssi 5836	. . . . . . . . . . . 12 |- ( ( A +o z ) e. B -> ( A +o z ) C_ B )
57	44, 56	syl 17	. . . . . . . . . . 11 |- ( z e. |^| S -> ( A +o z ) C_ B )
58	55, 57	mprgbir 2927	. . . . . . . . . 10 |- U_ z e. |^| S ( A +o z ) C_ B
59	54, 58	syl6eqss 3655	. . . . . . . . 9 |- ( ( |^| S e. _V /\ Lim |^| S ) -> ( A +o |^| S ) C_ B )
60	59	a1d 25	. . . . . . . 8 |- ( ( |^| S e. _V /\ Lim |^| S ) -> ( A C_ B -> ( A +o |^| S ) C_ B ) )
61	22, 52, 60	3jaoi 1391	. . . . . . 7 |- ( ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) -> ( A C_ B -> ( A +o |^| S ) C_ B ) )
62	16, 61	ax-mp 5	. . . . . 6 |- ( A C_ B -> ( A +o |^| S ) C_ B )
63	9	rspcev 3309	. . . . . . . . 9 |- ( ( B e. On /\ B C_ ( A +o B ) ) -> E. y e. On B C_ ( A +o y ) )
64	4, 7, 63	mp2an 708	. . . . . . . 8 |- E. y e. On B C_ ( A +o y )
65		nfcv 2764	. . . . . . . . . 10 |- F/_ y B
66		nfcv 2764	. . . . . . . . . . 11 |- F/_ y A
67		nfcv 2764	. . . . . . . . . . 11 |- F/_ y +o
68		nfrab1 3122	. . . . . . . . . . . 12 |- F/_ y { y e. On | B C_ ( A +o y ) }
69	68	nfint 4486	. . . . . . . . . . 11 |- F/_ y |^| { y e. On | B C_ ( A +o y ) }
70	66, 67, 69	nfov 6676	. . . . . . . . . 10 |- F/_ y ( A +o |^| { y e. On | B C_ ( A +o y ) } )
71	65, 70	nfss 3596	. . . . . . . . 9 |- F/ y B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } )
72		oveq2 6658	. . . . . . . . . 10 |- ( y = |^| { y e. On | B C_ ( A +o y ) } -> ( A +o y ) = ( A +o |^| { y e. On | B C_ ( A +o y ) } ) )
73	72	sseq2d 3633	. . . . . . . . 9 |- ( y = |^| { y e. On | B C_ ( A +o y ) } -> ( B C_ ( A +o y ) <-> B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } ) ) )
74	71, 73	onminsb 6999	. . . . . . . 8 |- ( E. y e. On B C_ ( A +o y ) -> B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } ) )
75	64, 74	ax-mp 5	. . . . . . 7 |- B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } )
76	32	oveq2i 6661	. . . . . . 7 |- ( A +o |^| S ) = ( A +o |^| { y e. On | B C_ ( A +o y ) } )
77	75, 76	sseqtr4i 3638	. . . . . 6 |- B C_ ( A +o |^| S )
78	62, 77	jctir 561	. . . . 5 |- ( A C_ B -> ( ( A +o |^| S ) C_ B /\ B C_ ( A +o |^| S ) ) )
79		eqss 3618	. . . . 5 |- ( ( A +o |^| S ) = B <-> ( ( A +o |^| S ) C_ B /\ B C_ ( A +o |^| S ) ) )
80	78, 79	sylibr 224	. . . 4 |- ( A C_ B -> ( A +o |^| S ) = B )
81		oveq2 6658	. . . . . 6 |- ( x = |^| S -> ( A +o x ) = ( A +o |^| S ) )
82	81	eqeq1d 2624	. . . . 5 |- ( x = |^| S -> ( ( A +o x ) = B <-> ( A +o |^| S ) = B ) )
83	82	rspcev 3309	. . . 4 |- ( ( |^| S e. On /\ ( A +o |^| S ) = B ) -> E. x e. On ( A +o x ) = B )
84	14, 80, 83	sylancr 695	. . 3 |- ( A C_ B -> E. x e. On ( A +o x ) = B )
85		eqtr3 2643	. . . . 5 |- ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> ( A +o x ) = ( A +o y ) )
86		oacan 7628	. . . . . 6 |- ( ( A e. On /\ x e. On /\ y e. On ) -> ( ( A +o x ) = ( A +o y ) <-> x = y ) )
87	5, 86	mp3an1 1411	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( ( A +o x ) = ( A +o y ) <-> x = y ) )
88	85, 87	syl5ib 234	. . . 4 |- ( ( x e. On /\ y e. On ) -> ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) )
89	88	rgen2a 2977	. . 3 |- A. x e. On A. y e. On ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y )
90	84, 89	jctir 561	. 2 |- ( A C_ B -> ( E. x e. On ( A +o x ) = B /\ A. x e. On A. y e. On ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) ) )
91		oveq2 6658	. . . 4 |- ( x = y -> ( A +o x ) = ( A +o y ) )
92	91	eqeq1d 2624	. . 3 |- ( x = y -> ( ( A +o x ) = B <-> ( A +o y ) = B ) )
93	92	reu4 3400	. 2 |- ( E! x e. On ( A +o x ) = B <-> ( E. x e. On ( A +o x ) = B /\ A. x e. On A. y e. On ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) ) )
94	90, 93	sylibr 224	1 |- ( A C_ B -> E! x e. On ( A +o x ) = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |^|cint 4475   U_ciun 4520   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725  (class class class)co 6650   +o coa 7557

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  oawordeu  7635