MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnsdomel Unicode version

Theorem nnsdomel 7812
Description: Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
nnsdomel  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  A 
~<  B ) )

Proof of Theorem nnsdomel
StepHypRef Expression
1 cardnn 7785 . . 3  |-  ( A  e.  om  ->  ( card `  A )  =  A )
2 cardnn 7785 . . 3  |-  ( B  e.  om  ->  ( card `  B )  =  B )
3 eleq12 2451 . . 3  |-  ( ( ( card `  A
)  =  A  /\  ( card `  B )  =  B )  ->  (
( card `  A )  e.  ( card `  B
)  <->  A  e.  B
) )
41, 2, 3syl2an 464 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( card `  A
)  e.  ( card `  B )  <->  A  e.  B ) )
5 nnon 4793 . . . 4  |-  ( A  e.  om  ->  A  e.  On )
6 onenon 7771 . . . 4  |-  ( A  e.  On  ->  A  e.  dom  card )
75, 6syl 16 . . 3  |-  ( A  e.  om  ->  A  e.  dom  card )
8 nnon 4793 . . . 4  |-  ( B  e.  om  ->  B  e.  On )
9 onenon 7771 . . . 4  |-  ( B  e.  On  ->  B  e.  dom  card )
108, 9syl 16 . . 3  |-  ( B  e.  om  ->  B  e.  dom  card )
11 cardsdom2 7810 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  e.  (
card `  B )  <->  A 
~<  B ) )
127, 10, 11syl2an 464 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( card `  A
)  e.  ( card `  B )  <->  A  ~<  B ) )
134, 12bitr3d 247 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  A 
~<  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4155   Oncon0 4524   omcom 4787   dom cdm 4820   ` cfv 5396    ~< csdm 7046   cardccrd 7757
This theorem is referenced by:  fin23lem27  8143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761
  Copyright terms: Public domain W3C validator