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Theorem oneo 6595
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 4500 . . 3  |-  ( A  e.  On  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
213ad2ant1 976 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o  .o  A ) )  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
3 suceq 4473 . . . . 5  |-  ( C  =  ( 2o  .o  A )  ->  suc  C  =  suc  ( 2o 
.o  A ) )
43eqeq1d 2304 . . . 4  |-  ( C  =  ( 2o  .o  A )  ->  ( suc  C  =  ( 2o 
.o  B )  <->  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) ) )
543ad2ant3 978 . . 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 5899 . . . . . . . 8  |-  ( 2o 
.o  A )  e. 
_V
76sucid 4487 . . . . . . 7  |-  ( 2o 
.o  A )  e. 
suc  ( 2o  .o  A )
8 eleq2 2357 . . . . . . 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 202 . . . . . 6  |-  ( suc  ( 2o  .o  A
)  =  ( 2o 
.o  B )  -> 
( 2o  .o  A
)  e.  ( 2o 
.o  B ) )
10 2on 6503 . . . . . . . 8  |-  2o  e.  On
11 omord 6582 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  2o  e.  On )  ->  (
( A  e.  B  /\  (/)  e.  2o )  <-> 
( 2o  .o  A
)  e.  ( 2o 
.o  B ) ) )
1210, 11mp3an3 1266 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  e.  B  /\  (/)  e.  2o ) 
<->  ( 2o  .o  A
)  e.  ( 2o 
.o  B ) ) )
13 simpl 443 . . . . . . 7  |-  ( ( A  e.  B  /\  (/) 
e.  2o )  ->  A  e.  B )
1412, 13syl6bir 220 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( 2o  .o  A )  e.  ( 2o  .o  B )  ->  A  e.  B
) )
159, 14syl5 28 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  ( 2o 
.o  A )  =  ( 2o  .o  B
)  ->  A  e.  B ) )
16 simpr 447 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  suc  ( 2o  .o  A
)  =  ( 2o 
.o  B ) )
17 omcl 6551 . . . . . . . . . . . . 13  |-  ( ( 2o  e.  On  /\  A  e.  On )  ->  ( 2o  .o  A
)  e.  On )
1810, 17mpan 651 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( 2o  .o  A )  e.  On )
19 oa1suc 6546 . . . . . . . . . . . 12  |-  ( ( 2o  .o  A )  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  =  suc  ( 2o 
.o  A ) )
2018, 19syl 15 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  =  suc  ( 2o 
.o  A ) )
21 1on 6502 . . . . . . . . . . . . . . . 16  |-  1o  e.  On
2221elexi 2810 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
2322sucid 4487 . . . . . . . . . . . . . 14  |-  1o  e.  suc  1o
24 df-2o 6496 . . . . . . . . . . . . . 14  |-  2o  =  suc  1o
2523, 24eleqtrri 2369 . . . . . . . . . . . . 13  |-  1o  e.  2o
26 oaord 6561 . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . 14  |-  ( ( 2o  .o  A )  e.  On  ->  ( 1o  e.  2o  <->  ( ( 2o  .o  A )  +o  1o )  e.  ( ( 2o  .o  A
)  +o  2o ) ) )
2818, 27syl 15 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( 1o  e.  2o  <->  ( ( 2o  .o  A )  +o  1o )  e.  ( ( 2o  .o  A
)  +o  2o ) ) )
2925, 28mpbii 202 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  e.  ( ( 2o 
.o  A )  +o  2o ) )
30 omsuc 6541 . . . . . . . . . . . . 13  |-  ( ( 2o  e.  On  /\  A  e.  On )  ->  ( 2o  .o  suc  A )  =  ( ( 2o  .o  A )  +o  2o ) )
3110, 30mpan 651 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( 2o  .o  suc  A )  =  ( ( 2o 
.o  A )  +o  2o ) )
3229, 31eleqtrrd 2373 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( 2o  .o  A
)  +o  1o )  e.  ( 2o  .o  suc  A ) )
3320, 32eqeltrrd 2371 . . . . . . . . . 10  |-  ( A  e.  On  ->  suc  ( 2o  .o  A
)  e.  ( 2o 
.o  suc  A )
)
3433ad2antrr 706 . . . . . . . . 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 2371 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  ( 2o  .o  B )  e.  ( 2o  .o  suc  A ) )
36 suceloni 4620 . . . . . . . . . . 11  |-  ( A  e.  On  ->  suc  A  e.  On )
37 omord 6582 . . . . . . . . . . . 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 1266 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  suc  A  e.  On )  ->  ( ( B  e.  suc  A  /\  (/) 
e.  2o )  <->  ( 2o  .o  B )  e.  ( 2o  .o  suc  A
) ) )
3936, 38sylan2 460 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( B  e. 
suc  A  /\  (/)  e.  2o ) 
<->  ( 2o  .o  B
)  e.  ( 2o 
.o  suc  A )
) )
4039ancoms 439 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( B  e. 
suc  A  /\  (/)  e.  2o ) 
<->  ( 2o  .o  B
)  e.  ( 2o 
.o  suc  A )
) )
4140adantr 451 . . . . . . . 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 223 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  ( B  e.  suc  A  /\  (/) 
e.  2o ) )
4342simpld 445 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  suc  ( 2o 
.o  A )  =  ( 2o  .o  B
) )  ->  B  e.  suc  A )
4443ex 423 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  ( 2o 
.o  A )  =  ( 2o  .o  B
)  ->  B  e.  suc  A ) )
4515, 44jcad 519 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  ( 2o 
.o  A )  =  ( 2o  .o  B
)  ->  ( A  e.  B  /\  B  e. 
suc  A ) ) )
46453adant3 975 . . 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 206 . 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 168 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   (/)c0 3468   Oncon0 4408   suc csuc 4410  (class class class)co 5874   1oc1o 6488   2oc2o 6489    +o coa 6492    .o comu 6493
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-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500
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