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Theorem fientri3 6732
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 6585 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 272 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6585 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 119 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad2antlr 476 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrr 506 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  A  ~~  n )
87adantr 272 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~~  n )
9 simpr 109 . . . . . . . 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 505 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  n  e.  om )
1110adantr 272 . . . . . . . . 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 505 . . . . . . . . 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 6687 . . . . . . . . 9  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  ~<_  m  <->  n  C_  m
) )
1411, 12, 13syl2anc 406 . . . . . . . 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 166 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  n  ~<_  m )
16 endomtr 6614 . . . . . . 7  |-  ( ( A  ~~  n  /\  n  ~<_  m )  ->  A  ~<_  m )
178, 15, 16syl2anc 406 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~<_  m )
18 simplrr 506 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  B  ~~  m )
1918ensymd 6607 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  m  ~~  B )
20 domentr 6615 . . . . . 6  |-  ( ( A  ~<_  m  /\  m  ~~  B )  ->  A  ~<_  B )
2117, 19, 20syl2anc 406 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~<_  B )
2221orcd 693 . . . 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 506 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~~  m )
24 simpr 109 . . . . . . . 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 505 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  m  e.  om )
2610adantr 272 . . . . . . . . 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 6687 . . . . . . . . 9  |-  ( ( m  e.  om  /\  n  e.  om )  ->  ( m  ~<_  n  <->  m  C_  n
) )
2825, 26, 27syl2anc 406 . . . . . . . 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 166 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  m  ~<_  n )
30 endomtr 6614 . . . . . . 7  |-  ( ( B  ~~  m  /\  m  ~<_  n )  ->  B  ~<_  n )
3123, 29, 30syl2anc 406 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~<_  n )
327adantr 272 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  A  ~~  n )
3332ensymd 6607 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  n  ~~  A )
34 domentr 6615 . . . . . 6  |-  ( ( B  ~<_  n  /\  n  ~~  A )  ->  B  ~<_  A )
3531, 33, 34syl2anc 406 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~<_  A )
3635olcd 694 . . . 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 501 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  m  e.  om )
38 nntri2or2 6324 . . . . 5  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  C_  m  \/  m  C_  n ) )
3910, 37, 38syl2anc 406 . . . 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 753 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
416, 40rexlimddv 2513 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
423, 41rexlimddv 2513 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670    e. wcel 1448   E.wrex 2376    C_ wss 3021   class class class wbr 3875   omcom 4442    ~~ cen 6562    ~<_ cdom 6563   Fincfn 6564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-er 6359  df-en 6565  df-dom 6566  df-fin 6567
This theorem is referenced by: (None)
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