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Theorem oneo 6816
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 4665 . . 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 4638 . . . . 5  |-  ( C  =  ( 2o  .o  A )  ->  suc  C  =  suc  ( 2o 
.o  A ) )
43eqeq1d 2443 . . . 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 6098 . . . . . . . 8  |-  ( 2o 
.o  A )  e. 
_V
76sucid 4652 . . . . . . 7  |-  ( 2o 
.o  A )  e. 
suc  ( 2o  .o  A )
8 eleq2 2496 . . . . . . 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 6724 . . . . . . . 8  |-  2o  e.  On
11 omord 6803 . . . . . . . 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 6772 . . . . . . . . . . . . 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 6767 . . . . . . . . . . . 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 6723 . . . . . . . . . . . . . . . 16  |-  1o  e.  On
2221elexi 2957 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
2322sucid 4652 . . . . . . . . . . . . . 14  |-  1o  e.  suc  1o
24 df-2o 6717 . . . . . . . . . . . . . 14  |-  2o  =  suc  1o
2523, 24eleqtrri 2508 . . . . . . . . . . . . 13  |-  1o  e.  2o
26 oaord 6782 . . . . . . . . . . . . . . 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 6762 . . . . . . . . . . . . 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 2512 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  e.  ( 2o  .o  suc  A ) )
3320, 32eqeltrrd 2510 . . . . . . . . . 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 2510 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  ( 2o  .o  B )  e.  ( 2o  .o  suc  A ) )
36 suceloni 4785 . . . . . . . . . . 11  |-  ( A  e.  On  ->  suc  A  e.  On )
37 omord 6803 . . . . . . . . . . . 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 1652    e. wcel 1725   (/)c0 3620   Oncon0 4573   suc csuc 4575  (class class class)co 6073   1oc1o 6709   2oc2o 6710    +o coa 6713    .o comu 6714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721
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