Theorem oeord 7668

Description: Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Assertion

Ref	Expression
oeord	|- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A e. B <-> ( C ^o A ) e. ( C ^o B ) ) )

Proof of Theorem oeord

Step	Hyp	Ref	Expression
1		oeordi 7667	. . 3 |- ( ( B e. On /\ C e. ( On \ 2o ) ) -> ( A e. B -> ( C ^o A ) e. ( C ^o B ) ) )
2	1	3adant1 1079	. 2 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A e. B -> ( C ^o A ) e. ( C ^o B ) ) )
3		oveq2 6658	. . . . . 6 |- ( A = B -> ( C ^o A ) = ( C ^o B ) )
4	3	a1i 11	. . . . 5 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A = B -> ( C ^o A ) = ( C ^o B ) ) )
5		oeordi 7667	. . . . . 6 |- ( ( A e. On /\ C e. ( On \ 2o ) ) -> ( B e. A -> ( C ^o B ) e. ( C ^o A ) ) )
6	5	3adant2 1080	. . . . 5 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( B e. A -> ( C ^o B ) e. ( C ^o A ) ) )
7	4, 6	orim12d 883	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( ( A = B \/ B e. A ) -> ( ( C ^o A ) = ( C ^o B ) \/ ( C ^o B ) e. ( C ^o A ) ) ) )
8	7	con3d 148	. . 3 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( -. ( ( C ^o A ) = ( C ^o B ) \/ ( C ^o B ) e. ( C ^o A ) ) -> -. ( A = B \/ B e. A ) ) )
9		eldifi 3732	. . . . . 6 |- ( C e. ( On \ 2o ) -> C e. On )
10	9	3ad2ant3 1084	. . . . 5 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> C e. On )
11		simp1 1061	. . . . 5 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> A e. On )
12		oecl 7617	. . . . 5 |- ( ( C e. On /\ A e. On ) -> ( C ^o A ) e. On )
13	10, 11, 12	syl2anc 693	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( C ^o A ) e. On )
14		simp2 1062	. . . . 5 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> B e. On )
15		oecl 7617	. . . . 5 |- ( ( C e. On /\ B e. On ) -> ( C ^o B ) e. On )
16	10, 14, 15	syl2anc 693	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( C ^o B ) e. On )
17		eloni 5733	. . . . 5 |- ( ( C ^o A ) e. On -> Ord ( C ^o A ) )
18		eloni 5733	. . . . 5 |- ( ( C ^o B ) e. On -> Ord ( C ^o B ) )
19		ordtri2 5758	. . . . 5 |- ( ( Ord ( C ^o A ) /\ Ord ( C ^o B ) ) -> ( ( C ^o A ) e. ( C ^o B ) <-> -. ( ( C ^o A ) = ( C ^o B ) \/ ( C ^o B ) e. ( C ^o A ) ) ) )
20	17, 18, 19	syl2an 494	. . . 4 |- ( ( ( C ^o A ) e. On /\ ( C ^o B ) e. On ) -> ( ( C ^o A ) e. ( C ^o B ) <-> -. ( ( C ^o A ) = ( C ^o B ) \/ ( C ^o B ) e. ( C ^o A ) ) ) )
21	13, 16, 20	syl2anc 693	. . 3 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( ( C ^o A ) e. ( C ^o B ) <-> -. ( ( C ^o A ) = ( C ^o B ) \/ ( C ^o B ) e. ( C ^o A ) ) ) )
22		eloni 5733	. . . . 5 |- ( A e. On -> Ord A )
23		eloni 5733	. . . . 5 |- ( B e. On -> Ord B )
24		ordtri2 5758	. . . . 5 |- ( ( Ord A /\ Ord B ) -> ( A e. B <-> -. ( A = B \/ B e. A ) ) )
25	22, 23, 24	syl2an 494	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A e. B <-> -. ( A = B \/ B e. A ) ) )
26	25	3adant3 1081	. . 3 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A e. B <-> -. ( A = B \/ B e. A ) ) )
27	8, 21, 26	3imtr4d 283	. 2 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( ( C ^o A ) e. ( C ^o B ) -> A e. B ) )
28	2, 27	impbid 202	1 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A e. B <-> ( C ^o A ) e. ( C ^o B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   Ord word 5722   Oncon0 5723  (class class class)co 6650   2oc2o 7554   ^o coe 7559

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-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-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-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oeword  7670  oeeui  7682  omabs  7727  cantnflem3  8588