ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fientri3 Unicode version

Theorem fientri3 6353
Description: Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
Assertion
Ref Expression
fientri3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )

Proof of Theorem fientri3
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6241 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 113 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 261 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6241 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 113 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad2antlr 458 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrr 488 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  A  ~~  n )
87adantr 261 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~~  n )
9 simpr 103 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  n  C_  m )
10 simplrl 487 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  n  e.  om )
1110adantr 261 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  n  e.  om )
12 simplrl 487 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  m  e.  om )
13 nndomo 6326 . . . . . . . . 9  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  ~<_  m  <->  n  C_  m
) )
1411, 12, 13syl2anc 391 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  -> 
( n  ~<_  m  <->  n  C_  m
) )
159, 14mpbird 156 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  n  ~<_  m )
16 endomtr 6270 . . . . . . 7  |-  ( ( A  ~~  n  /\  n  ~<_  m )  ->  A  ~<_  m )
178, 15, 16syl2anc 391 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~<_  m )
18 simplrr 488 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  B  ~~  m )
1918ensymd 6263 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  m  ~~  B )
20 domentr 6271 . . . . . 6  |-  ( ( A  ~<_  m  /\  m  ~~  B )  ->  A  ~<_  B )
2117, 19, 20syl2anc 391 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~<_  B )
2221orcd 652 . . . 4  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
23 simplrr 488 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~~  m )
24 simpr 103 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  m  C_  n )
25 simplrl 487 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  m  e.  om )
2610adantr 261 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  n  e.  om )
27 nndomo 6326 . . . . . . . . 9  |-  ( ( m  e.  om  /\  n  e.  om )  ->  ( m  ~<_  n  <->  m  C_  n
) )
2825, 26, 27syl2anc 391 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  -> 
( m  ~<_  n  <->  m  C_  n
) )
2924, 28mpbird 156 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  m  ~<_  n )
30 endomtr 6270 . . . . . . 7  |-  ( ( B  ~~  m  /\  m  ~<_  n )  ->  B  ~<_  n )
3123, 29, 30syl2anc 391 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~<_  n )
327adantr 261 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  A  ~~  n )
3332ensymd 6263 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  n  ~~  A )
34 domentr 6271 . . . . . 6  |-  ( ( B  ~<_  n  /\  n  ~~  A )  ->  B  ~<_  A )
3531, 33, 34syl2anc 391 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~<_  A )
3635olcd 653 . . . 4  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
37 simprl 483 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  m  e.  om )
38 nntri2or2 6076 . . . . 5  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  C_  m  \/  m  C_  n ) )
3910, 37, 38syl2anc 391 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( n  C_  m  \/  m  C_  n ) )
4022, 36, 39mpjaodan 711 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
416, 40rexlimddv 2437 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
423, 41rexlimddv 2437 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    e. wcel 1393   E.wrex 2307    C_ wss 2917   class class class wbr 3764   omcom 4313    ~~ cen 6219    ~<_ cdom 6220   Fincfn 6221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-er 6106  df-en 6222  df-dom 6223  df-fin 6224
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator