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Theorem oeword 6825
Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oeword  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )

Proof of Theorem oeword
StepHypRef Expression
1 oeord 6823 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
2 oecan 6824 . . . . 5  |-  ( ( C  e.  ( On 
\  2o )  /\  A  e.  On  /\  B  e.  On )  ->  (
( C  ^o  A
)  =  ( C  ^o  B )  <->  A  =  B ) )
323coml 1160 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  =  ( C  ^o  B )  <->  A  =  B ) )
43bicomd 193 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  =  B  <->  ( C  ^o  A )  =  ( C  ^o  B ) ) )
51, 4orbi12d 691 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( A  e.  B  \/  A  =  B
)  <->  ( ( C  ^o  A )  e.  ( C  ^o  B
)  \/  ( C  ^o  A )  =  ( C  ^o  B
) ) ) )
6 onsseleq 4614 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
763adant3 977 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
8 eldifi 3461 . . . 4  |-  ( C  e.  ( On  \  2o )  ->  C  e.  On )
9 id 20 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  On  /\  B  e.  On ) )
10 oecl 6773 . . . . . 6  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
11 oecl 6773 . . . . . 6  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1210, 11anim12dan 811 . . . . 5  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  ^o  A )  e.  On  /\  ( C  ^o  B )  e.  On ) )
13 onsseleq 4614 . . . . 5  |-  ( ( ( C  ^o  A
)  e.  On  /\  ( C  ^o  B )  e.  On )  -> 
( ( C  ^o  A )  C_  ( C  ^o  B )  <->  ( ( C  ^o  A )  e.  ( C  ^o  B
)  \/  ( C  ^o  A )  =  ( C  ^o  B
) ) ) )
1412, 13syl 16 . . . 4  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  ^o  A )  C_  ( C  ^o  B )  <-> 
( ( C  ^o  A )  e.  ( C  ^o  B )  \/  ( C  ^o  A )  =  ( C  ^o  B ) ) ) )
158, 9, 14syl2anr 465 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  ( On  \  2o ) )  ->  ( ( C  ^o  A )  C_  ( C  ^o  B )  <-> 
( ( C  ^o  A )  e.  ( C  ^o  B )  \/  ( C  ^o  A )  =  ( C  ^o  B ) ) ) )
16153impa 1148 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  C_  ( C  ^o  B )  <->  ( ( C  ^o  A )  e.  ( C  ^o  B
)  \/  ( C  ^o  A )  =  ( C  ^o  B
) ) ) )
175, 7, 163bitr4d 277 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309    C_ wss 3312   Oncon0 4573  (class class class)co 6073   2oc2o 6710    ^o coe 6715
This theorem is referenced by:  oewordi  6826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-oexp 6722
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