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Theorem oneo 6783
Description: If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)
Assertion
Ref Expression
oneo  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o  .o  A ) )  ->  -.  suc  C  =  ( 2o  .o  B ) )

Proof of Theorem oneo
StepHypRef Expression
1 onnbtwn 4632 . . 3  |-  ( A  e.  On  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
213ad2ant1 978 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o  .o  A ) )  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
3 suceq 4606 . . . . 5  |-  ( C  =  ( 2o  .o  A )  ->  suc  C  =  suc  ( 2o 
.o  A ) )
43eqeq1d 2412 . . . 4  |-  ( C  =  ( 2o  .o  A )  ->  ( suc  C  =  ( 2o 
.o  B )  <->  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) ) )
543ad2ant3 980 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o  .o  A ) )  -> 
( suc  C  =  ( 2o  .o  B
)  <->  suc  ( 2o  .o  A )  =  ( 2o  .o  B ) ) )
6 ovex 6065 . . . . . . . 8  |-  ( 2o 
.o  A )  e. 
_V
76sucid 4620 . . . . . . 7  |-  ( 2o 
.o  A )  e. 
suc  ( 2o  .o  A )
8 eleq2 2465 . . . . . . 7  |-  ( suc  ( 2o  .o  A
)  =  ( 2o 
.o  B )  -> 
( ( 2o  .o  A )  e.  suc  ( 2o  .o  A
)  <->  ( 2o  .o  A )  e.  ( 2o  .o  B ) ) )
97, 8mpbii 203 . . . . . 6  |-  ( suc  ( 2o  .o  A
)  =  ( 2o 
.o  B )  -> 
( 2o  .o  A
)  e.  ( 2o 
.o  B ) )
10 2on 6691 . . . . . . . 8  |-  2o  e.  On
11 omord 6770 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  2o  e.  On )  ->  (
( A  e.  B  /\  (/)  e.  2o )  <-> 
( 2o  .o  A
)  e.  ( 2o 
.o  B ) ) )
1210, 11mp3an3 1268 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  e.  B  /\  (/)  e.  2o ) 
<->  ( 2o  .o  A
)  e.  ( 2o 
.o  B ) ) )
13 simpl 444 . . . . . . 7  |-  ( ( A  e.  B  /\  (/) 
e.  2o )  ->  A  e.  B )
1412, 13syl6bir 221 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( 2o  .o  A )  e.  ( 2o  .o  B )  ->  A  e.  B
) )
159, 14syl5 30 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  ( 2o 
.o  A )  =  ( 2o  .o  B
)  ->  A  e.  B ) )
16 simpr 448 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  suc  ( 2o  .o  A
)  =  ( 2o 
.o  B ) )
17 omcl 6739 . . . . . . . . . . . . 13  |-  ( ( 2o  e.  On  /\  A  e.  On )  ->  ( 2o  .o  A
)  e.  On )
1810, 17mpan 652 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( 2o  .o  A )  e.  On )
19 oa1suc 6734 . . . . . . . . . . . 12  |-  ( ( 2o  .o  A )  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  =  suc  ( 2o 
.o  A ) )
2018, 19syl 16 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  =  suc  ( 2o 
.o  A ) )
21 1on 6690 . . . . . . . . . . . . . . . 16  |-  1o  e.  On
2221elexi 2925 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
2322sucid 4620 . . . . . . . . . . . . . 14  |-  1o  e.  suc  1o
24 df-2o 6684 . . . . . . . . . . . . . 14  |-  2o  =  suc  1o
2523, 24eleqtrri 2477 . . . . . . . . . . . . 13  |-  1o  e.  2o
26 oaord 6749 . . . . . . . . . . . . . . 15  |-  ( ( 1o  e.  On  /\  2o  e.  On  /\  ( 2o  .o  A )  e.  On )  ->  ( 1o  e.  2o  <->  ( ( 2o  .o  A )  +o  1o )  e.  ( ( 2o  .o  A
)  +o  2o ) ) )
2721, 10, 26mp3an12 1269 . . . . . . . . . . . . . 14  |-  ( ( 2o  .o  A )  e.  On  ->  ( 1o  e.  2o  <->  ( ( 2o  .o  A )  +o  1o )  e.  ( ( 2o  .o  A
)  +o  2o ) ) )
2818, 27syl 16 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( 1o  e.  2o  <->  ( ( 2o  .o  A )  +o  1o )  e.  ( ( 2o  .o  A
)  +o  2o ) ) )
2925, 28mpbii 203 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  e.  ( ( 2o 
.o  A )  +o  2o ) )
30 omsuc 6729 . . . . . . . . . . . . 13  |-  ( ( 2o  e.  On  /\  A  e.  On )  ->  ( 2o  .o  suc  A )  =  ( ( 2o  .o  A )  +o  2o ) )
3110, 30mpan 652 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( 2o  .o  suc  A )  =  ( ( 2o 
.o  A )  +o  2o ) )
3229, 31eleqtrrd 2481 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  e.  ( 2o  .o  suc  A ) )
3320, 32eqeltrrd 2479 . . . . . . . . . 10  |-  ( A  e.  On  ->  suc  ( 2o  .o  A
)  e.  ( 2o 
.o  suc  A )
)
3433ad2antrr 707 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  suc  ( 2o  .o  A
)  e.  ( 2o 
.o  suc  A )
)
3516, 34eqeltrrd 2479 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  ( 2o  .o  B )  e.  ( 2o  .o  suc  A ) )
36 suceloni 4752 . . . . . . . . . . 11  |-  ( A  e.  On  ->  suc  A  e.  On )
37 omord 6770 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  suc  A  e.  On  /\  2o  e.  On )  -> 
( ( B  e. 
suc  A  /\  (/)  e.  2o ) 
<->  ( 2o  .o  B
)  e.  ( 2o 
.o  suc  A )
) )
3810, 37mp3an3 1268 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  suc  A  e.  On )  ->  ( ( B  e.  suc  A  /\  (/) 
e.  2o )  <->  ( 2o  .o  B )  e.  ( 2o  .o  suc  A
) ) )
3936, 38sylan2 461 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( B  e. 
suc  A  /\  (/)  e.  2o ) 
<->  ( 2o  .o  B
)  e.  ( 2o 
.o  suc  A )
) )
4039ancoms 440 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( B  e. 
suc  A  /\  (/)  e.  2o ) 
<->  ( 2o  .o  B
)  e.  ( 2o 
.o  suc  A )
) )
4140adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  (
( B  e.  suc  A  /\  (/)  e.  2o )  <-> 
( 2o  .o  B
)  e.  ( 2o 
.o  suc  A )
) )
4235, 41mpbird 224 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  ( B  e.  suc  A  /\  (/) 
e.  2o ) )
4342simpld 446 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  B  e.  suc  A )
4443ex 424 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  ( 2o 
.o  A )  =  ( 2o  .o  B
)  ->  B  e.  suc  A ) )
4515, 44jcad 520 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  ( 2o 
.o  A )  =  ( 2o  .o  B
)  ->  ( A  e.  B  /\  B  e. 
suc  A ) ) )
46453adant3 977 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o  .o  A ) )  -> 
( suc  ( 2o  .o  A )  =  ( 2o  .o  B )  ->  ( A  e.  B  /\  B  e. 
suc  A ) ) )
475, 46sylbid 207 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o  .o  A ) )  -> 
( suc  C  =  ( 2o  .o  B
)  ->  ( A  e.  B  /\  B  e. 
suc  A ) ) )
482, 47mtod 170 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o  .o  A ) )  ->  -.  suc  C  =  ( 2o  .o  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   (/)c0 3588   Oncon0 4541   suc csuc 4543  (class class class)co 6040   1oc1o 6676   2oc2o 6677    +o coa 6680    .o comu 6681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688
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