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Theorem omsucdomOLD 7072
Description: Strict dominance of natural numbers is the same as dominance over the successor of the smaller. (Contributed by NM, 25-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
omsucdomOLD  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )

Proof of Theorem omsucdomOLD
StepHypRef Expression
1 nnord 4680 . . 3  |-  ( B  e.  om  ->  Ord  B )
2 nnord 4680 . . . . 5  |-  ( A  e.  om  ->  Ord  A )
3 ordelpss 4436 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
42, 3sylan 457 . . . 4  |-  ( ( A  e.  om  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
5 ordelsuc 4627 . . . 4  |-  ( ( A  e.  om  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
64, 5bitr3d 246 . . 3  |-  ( ( A  e.  om  /\  Ord  B )  ->  ( A  C.  B  <->  suc  A  C_  B ) )
71, 6sylan2 460 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C.  B  <->  suc 
A  C_  B )
)
8 nnsdomo 7071 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~<  B  <->  A  C.  B ) )
9 peano2b 4688 . . 3  |-  ( A  e.  om  <->  suc  A  e. 
om )
10 nndomo 7070 . . 3  |-  ( ( suc  A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~<_  B  <->  suc  A  C_  B )
)
119, 10sylanb 458 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~<_  B  <->  suc  A  C_  B )
)
127, 8, 113bitr4d 276 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696    C_ wss 3165    C. wpss 3166   class class class wbr 4039   Ord word 4407   suc csuc 4410   omcom 4672    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  fisucdomOLD  7082
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-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-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-opab 4094  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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