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Theorem fientri3 6771
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 6623 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 274 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6623 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 119 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad2antlr 480 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrr 510 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  A  ~~  n )
87adantr 274 . . . . . . 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 509 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  n  e.  om )
1110adantr 274 . . . . . . . . 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 509 . . . . . . . . 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 6726 . . . . . . . . 9  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  ~<_  m  <->  n  C_  m
) )
1411, 12, 13syl2anc 408 . . . . . . . 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 6652 . . . . . . 7  |-  ( ( A  ~~  n  /\  n  ~<_  m )  ->  A  ~<_  m )
178, 15, 16syl2anc 408 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~<_  m )
18 simplrr 510 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  B  ~~  m )
1918ensymd 6645 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  m  ~~  B )
20 domentr 6653 . . . . . 6  |-  ( ( A  ~<_  m  /\  m  ~~  B )  ->  A  ~<_  B )
2117, 19, 20syl2anc 408 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  n  C_  m )  ->  A  ~<_  B )
2221orcd 707 . . . 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 510 . . . . . . 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 509 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  m  e.  om )
2610adantr 274 . . . . . . . . 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 6726 . . . . . . . . 9  |-  ( ( m  e.  om  /\  n  e.  om )  ->  ( m  ~<_  n  <->  m  C_  n
) )
2825, 26, 27syl2anc 408 . . . . . . . 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 6652 . . . . . . 7  |-  ( ( B  ~~  m  /\  m  ~<_  n )  ->  B  ~<_  n )
3123, 29, 30syl2anc 408 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~<_  n )
327adantr 274 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  A  ~~  n )
3332ensymd 6645 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  n  ~~  A )
34 domentr 6653 . . . . . 6  |-  ( ( B  ~<_  n  /\  n  ~~  A )  ->  B  ~<_  A )
3531, 33, 34syl2anc 408 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  /\  m  C_  n )  ->  B  ~<_  A )
3635olcd 708 . . . 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 505 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  m  e.  om )
38 nntri2or2 6362 . . . . 5  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  C_  m  \/  m  C_  n ) )
3910, 37, 38syl2anc 408 . . . 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 772 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
416, 40rexlimddv 2531 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
423, 41rexlimddv 2531 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 682    e. wcel 1465   E.wrex 2394    C_ wss 3041   class class class wbr 3899   omcom 4474    ~~ cen 6600    ~<_ cdom 6601   Fincfn 6602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-er 6397  df-en 6603  df-dom 6604  df-fin 6605
This theorem is referenced by: (None)
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