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Theorem oeword 7241
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 7239 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
2 oecan 7240 . . . . 5  |-  ( ( C  e.  ( On 
\  2o )  /\  A  e.  On  /\  B  e.  On )  ->  (
( C  ^o  A
)  =  ( C  ^o  B )  <->  A  =  B ) )
323coml 1204 . . . 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 4909 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
763adant3 1017 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
8 eldifi 3611 . . . 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 7189 . . . . . 6  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
11 oecl 7189 . . . . . 6  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1210, 11anim12dan 837 . . . . 5  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  ^o  A )  e.  On  /\  ( C  ^o  B )  e.  On ) )
13 onsseleq 4909 . . . . 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 1192 . 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 974    = wceq 1383    e. wcel 1804    \ cdif 3458    C_ wss 3461   Oncon0 4868  (class class class)co 6281   2oc2o 7126    ^o coe 7131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138
This theorem is referenced by:  oewordi  7242
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