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Theorem oawordriexmid 6134
Description: A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6133. (Contributed by Jim Kingdon, 15-May-2022.)
Hypothesis
Ref Expression
oawordriexmid.1  |-  ( ( a  e.  On  /\  b  e.  On  /\  c  e.  On )  ->  (
a  C_  b  ->  ( a  +o  c ) 
C_  ( b  +o  c ) ) )
Assertion
Ref Expression
oawordriexmid  |-  ( ph  \/  -.  ph )
Distinct variable groups:    a, b, c    ph, a
Allowed substitution hints:    ph( b, c)

Proof of Theorem oawordriexmid
StepHypRef Expression
1 1on 6092 . . . . 5  |-  1o  e.  On
2 oawordriexmid.1 . . . . . . . 8  |-  ( ( a  e.  On  /\  b  e.  On  /\  c  e.  On )  ->  (
a  C_  b  ->  ( a  +o  c ) 
C_  ( b  +o  c ) ) )
323expa 1139 . . . . . . 7  |-  ( ( ( a  e.  On  /\  b  e.  On )  /\  c  e.  On )  ->  ( a  C_  b  ->  ( a  +o  c )  C_  (
b  +o  c ) ) )
43expcom 114 . . . . . 6  |-  ( c  e.  On  ->  (
( a  e.  On  /\  b  e.  On )  ->  ( a  C_  b  ->  ( a  +o  c )  C_  (
b  +o  c ) ) ) )
54rgen 2421 . . . . 5  |-  A. c  e.  On  ( ( a  e.  On  /\  b  e.  On )  ->  (
a  C_  b  ->  ( a  +o  c ) 
C_  ( b  +o  c ) ) )
6 oveq2 5571 . . . . . . . . 9  |-  ( c  =  1o  ->  (
a  +o  c )  =  ( a  +o  1o ) )
7 oveq2 5571 . . . . . . . . 9  |-  ( c  =  1o  ->  (
b  +o  c )  =  ( b  +o  1o ) )
86, 7sseq12d 3037 . . . . . . . 8  |-  ( c  =  1o  ->  (
( a  +o  c
)  C_  ( b  +o  c )  <->  ( a  +o  1o )  C_  (
b  +o  1o ) ) )
98imbi2d 228 . . . . . . 7  |-  ( c  =  1o  ->  (
( a  C_  b  ->  ( a  +o  c
)  C_  ( b  +o  c ) )  <->  ( a  C_  b  ->  ( a  +o  1o )  C_  (
b  +o  1o ) ) ) )
109imbi2d 228 . . . . . 6  |-  ( c  =  1o  ->  (
( ( a  e.  On  /\  b  e.  On )  ->  (
a  C_  b  ->  ( a  +o  c ) 
C_  ( b  +o  c ) ) )  <-> 
( ( a  e.  On  /\  b  e.  On )  ->  (
a  C_  b  ->  ( a  +o  1o ) 
C_  ( b  +o  1o ) ) ) ) )
1110rspcv 2706 . . . . 5  |-  ( 1o  e.  On  ->  ( A. c  e.  On  ( ( a  e.  On  /\  b  e.  On )  ->  (
a  C_  b  ->  ( a  +o  c ) 
C_  ( b  +o  c ) ) )  ->  ( ( a  e.  On  /\  b  e.  On )  ->  (
a  C_  b  ->  ( a  +o  1o ) 
C_  ( b  +o  1o ) ) ) ) )
121, 5, 11mp2 16 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  C_  b  ->  ( a  +o  1o )  C_  ( b  +o  1o ) ) )
13 oa1suc 6131 . . . . . 6  |-  ( a  e.  On  ->  (
a  +o  1o )  =  suc  a )
1413adantr 270 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  +o  1o )  =  suc  a )
15 oa1suc 6131 . . . . . 6  |-  ( b  e.  On  ->  (
b  +o  1o )  =  suc  b )
1615adantl 271 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( b  +o  1o )  =  suc  b )
1714, 16sseq12d 3037 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( a  +o  1o )  C_  (
b  +o  1o )  <->  suc  a  C_  suc  b
) )
1812, 17sylibd 147 . . 3  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  C_  b  ->  suc  a  C_  suc  b ) )
1918rgen2a 2422 . 2  |-  A. a  e.  On  A. b  e.  On  ( a  C_  b  ->  suc  a  C_  suc  b )
2019onsucsssucexmid 4298 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353    C_ wss 2982   Oncon0 4146   suc csuc 4148  (class class class)co 5563   1oc1o 6078    +o coa 6082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-oadd 6089
This theorem is referenced by: (None)
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