Theorem oewordi 7130

Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)

Assertion

Ref	Expression
oewordi	|- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) )

Proof of Theorem oewordi

Step	Hyp	Ref	Expression
1		eloni 4827	. . . . . 6 |- ( C e. On -> Ord C )
2		ordgt0ge1 7037	. . . . . 6 |- ( Ord C -> ( (/) e. C <-> 1o C_ C ) )
3	1, 2	syl 16	. . . . 5 |- ( C e. On -> ( (/) e. C <-> 1o C_ C ) )
4		1on 7027	. . . . . 6 |- 1o e. On
5		onsseleq 4858	. . . . . 6 |- ( ( 1o e. On /\ C e. On ) -> ( 1o C_ C <-> ( 1o e. C \/ 1o = C ) ) )
6	4, 5	mpan 670	. . . . 5 |- ( C e. On -> ( 1o C_ C <-> ( 1o e. C \/ 1o = C ) ) )
7	3, 6	bitrd 253	. . . 4 |- ( C e. On -> ( (/) e. C <-> ( 1o e. C \/ 1o = C ) ) )
8	7	3ad2ant3 1011	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( (/) e. C <-> ( 1o e. C \/ 1o = C ) ) )
9		ondif2 7042	. . . . . . 7 |- ( C e. ( On \ 2o ) <-> ( C e. On /\ 1o e. C ) )
10		oeword 7129	. . . . . . . . 9 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A C_ B <-> ( C ^o A ) C_ ( C ^o B ) ) )
11	10	biimpd 207	. . . . . . . 8 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) )
12	11	3expia 1190	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( C e. ( On \ 2o ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
13	9, 12	syl5bir 218	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( C e. On /\ 1o e. C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
14	13	expd 436	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( C e. On -> ( 1o e. C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) ) )
15	14	3impia 1185	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( 1o e. C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
16		oe1m 7084	. . . . . . . . . 10 |- ( A e. On -> ( 1o ^o A ) = 1o )
17	16	adantr 465	. . . . . . . . 9 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o A ) = 1o )
18		oe1m 7084	. . . . . . . . . 10 |- ( B e. On -> ( 1o ^o B ) = 1o )
19	18	adantl 466	. . . . . . . . 9 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o B ) = 1o )
20	17, 19	eqtr4d 2495	. . . . . . . 8 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o A ) = ( 1o ^o B ) )
21		eqimss 3506	. . . . . . . 8 |- ( ( 1o ^o A ) = ( 1o ^o B ) -> ( 1o ^o A ) C_ ( 1o ^o B ) )
22	20, 21	syl 16	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o A ) C_ ( 1o ^o B ) )
23		oveq1 6197	. . . . . . . 8 |- ( 1o = C -> ( 1o ^o A ) = ( C ^o A ) )
24		oveq1 6197	. . . . . . . 8 |- ( 1o = C -> ( 1o ^o B ) = ( C ^o B ) )
25	23, 24	sseq12d 3483	. . . . . . 7 |- ( 1o = C -> ( ( 1o ^o A ) C_ ( 1o ^o B ) <-> ( C ^o A ) C_ ( C ^o B ) ) )
26	22, 25	syl5ibcom 220	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( 1o = C -> ( C ^o A ) C_ ( C ^o B ) ) )
27	26	3adant3 1008	. . . . 5 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( 1o = C -> ( C ^o A ) C_ ( C ^o B ) ) )
28	27	a1dd 46	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( 1o = C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
29	15, 28	jaod 380	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( 1o e. C \/ 1o = C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
30	8, 29	sylbid 215	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( (/) e. C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
31	30	imp 429	1 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   \ cdif 3423   C_ wss 3426   (/)c0 3735   Ord word 4816   Oncon0 4817  (class class class)co 6190   1oc1o 7013   2oc2o 7014   ^o coe 7019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-omul 7025  df-oexp 7026

This theorem is referenced by:  oelim2  7134  oeoalem  7135  oeoelem  7137  oaabs2  7184  cantnflt  7981  cantnfltOLD  8011  cnfcom  8034  cnfcomOLD  8042