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Theorem sucdom 7201
Description: Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.)
Assertion
Ref Expression
sucdom  |-  ( A  e.  om  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )

Proof of Theorem sucdom
StepHypRef Expression
1 sucdom2 7200 . 2  |-  ( A 
~<  B  ->  suc  A  ~<_  B )
2 php4 7191 . . 3  |-  ( A  e.  om  ->  A  ~<  suc  A )
3 sdomdomtr 7137 . . . 4  |-  ( ( A  ~<  suc  A  /\  suc  A  ~<_  B )  ->  A  ~<  B )
43ex 423 . . 3  |-  ( A 
~<  suc  A  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )
52, 4syl 15 . 2  |-  ( A  e.  om  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )
61, 5impbid2 195 1  |-  ( A  e.  om  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1715   class class class wbr 4125   suc csuc 4497   omcom 4759    ~<_ cdom 7004    ~< csdm 7005
This theorem is referenced by:  0sdom1dom  7203  1sdom  7208  isnzr2  16225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-1o 6621  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009
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