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Theorem oeword 6604
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 6602 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
2 oecan 6603 . . . . 5  |-  ( ( C  e.  ( On 
\  2o )  /\  A  e.  On  /\  B  e.  On )  ->  (
( C  ^o  A
)  =  ( C  ^o  B )  <->  A  =  B ) )
323coml 1158 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  =  ( C  ^o  B )  <->  A  =  B ) )
43bicomd 192 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  =  B  <->  ( C  ^o  A )  =  ( C  ^o  B ) ) )
51, 4orbi12d 690 . 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 4449 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
763adant3 975 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
8 eldifi 3311 . . . 4  |-  ( C  e.  ( On  \  2o )  ->  C  e.  On )
9 id 19 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  On  /\  B  e.  On ) )
10 oecl 6552 . . . . . 6  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
11 oecl 6552 . . . . . 6  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1210, 11anim12dan 810 . . . . 5  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  ^o  A )  e.  On  /\  ( C  ^o  B )  e.  On ) )
13 onsseleq 4449 . . . . 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 15 . . . 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 464 . . 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 1146 . 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 276 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   Oncon0 4408  (class class class)co 5874   2oc2o 6489    ^o coe 6494
This theorem is referenced by:  oewordi  6605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501
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