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Theorem oeword 7229
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 7227 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
2 oecan 7228 . . . . 5  |-  ( ( C  e.  ( On 
\  2o )  /\  A  e.  On  /\  B  e.  On )  ->  (
( C  ^o  A
)  =  ( C  ^o  B )  <->  A  =  B ) )
323coml 1198 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  =  ( C  ^o  B )  <->  A  =  B ) )
43bicomd 201 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  =  B  <->  ( C  ^o  A )  =  ( C  ^o  B ) ) )
51, 4orbi12d 709 . 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 4912 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
763adant3 1011 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
8 eldifi 3619 . . . 4  |-  ( C  e.  ( On  \  2o )  ->  C  e.  On )
9 id 22 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  On  /\  B  e.  On ) )
10 oecl 7177 . . . . . 6  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
11 oecl 7177 . . . . . 6  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1210, 11anim12dan 834 . . . . 5  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  ^o  A )  e.  On  /\  ( C  ^o  B )  e.  On ) )
13 onsseleq 4912 . . . . 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 478 . . 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 1186 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    \ cdif 3466    C_ wss 3469   Oncon0 4871  (class class class)co 6275   2oc2o 7114    ^o coe 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-oexp 7126
This theorem is referenced by:  oewordi  7230
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