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Theorem sucdomiOLD 7144
Description: Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 7143. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sucdomiOLD  |-  ( ( A  e.  om  /\  B  e.  C )  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )

Proof of Theorem sucdomiOLD
StepHypRef Expression
1 php4 7133 . . 3  |-  ( A  e.  om  ->  A  ~<  suc  A )
2 sdomdomtr 7079 . . . 4  |-  ( ( A  ~<  suc  A  /\  suc  A  ~<_  B )  ->  A  ~<  B )
32ex 423 . . 3  |-  ( A 
~<  suc  A  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )
41, 3syl 15 . 2  |-  ( A  e.  om  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )
54adantr 451 1  |-  ( ( A  e.  om  /\  B  e.  C )  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   class class class wbr 4102   suc csuc 4473   omcom 4735    ~<_ cdom 6946    ~< csdm 6947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951
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