ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fientri3 Unicode version
Theorem fientri3 6944
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 6788 . . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
21biimpi 120 . . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
32adantr 276 . 2 |- ( ( A e. Fin /\ B e. Fin ) -> E. n e. om A ~~ n )
4 isfi 6788 . . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
54biimpi 120 . . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
65ad2antlr 489 . . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7 simplrr 536 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ n )
87adantr 276 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~~ n )
9 simpr 110 . . . . . . . 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 535 . . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n e. om )
1110adantr 276 . . . . . . . . 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 535 . . . . . . . . 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 6893 . . . . . . . . 9 |- ( ( n e. om /\ m e. om ) -> ( n ~<_ m <-> n C_ m ) )
1411, 12, 13syl2anc 411 . . . . . . . 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 167 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> n ~<_ m )
16 endomtr 6817 . . . . . . 7 |- ( ( A ~~ n /\ n ~<_ m ) -> A ~<_ m )
178, 15, 16syl2anc 411 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~<_ m )
18 simplrr 536 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> B ~~ m )
1918ensymd 6810 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> m ~~ B )
20 domentr 6818 . . . . . 6 |- ( ( A ~<_ m /\ m ~~ B ) -> A ~<_ B )
2117, 19, 20syl2anc 411 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~<_ B )
2221orcd 734 . . . 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 536 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> B ~~ m )
24 simpr 110 . . . . . . . 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 535 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> m e. om )
2610adantr 276 . . . . . . . . 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 6893 . . . . . . . . 9 |- ( ( m e. om /\ n e. om ) -> ( m ~<_ n <-> m C_ n ) )
2825, 26, 27syl2anc 411 . . . . . . . 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 167 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> m ~<_ n )
30 endomtr 6817 . . . . . . 7 |- ( ( B ~~ m /\ m ~<_ n ) -> B ~<_ n )
3123, 29, 30syl2anc 411 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> B ~<_ n )
327adantr 276 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> A ~~ n )
3332ensymd 6810 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> n ~~ A )
34 domentr 6818 . . . . . 6 |- ( ( B ~<_ n /\ n ~~ A ) -> B ~<_ A )
3531, 33, 34syl2anc 411 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> B ~<_ A )
3635olcd 735 . . . 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 529 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
38 nntri2or2 6524 . . . . 5 |- ( ( n e. om /\ m e. om ) -> ( n C_ m \/ m C_ n ) )
3910, 37, 38syl2anc 411 . . . 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 799 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ B \/ B ~<_ A ) )
416, 40rexlimddv 2612 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( A ~<_ B \/ B ~<_ A ) )
423, 41rexlimddv 2612 1 |- ( ( A e. Fin /\ B e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   e. wcel 2160   E.wrex 2469   C_ wss 3144   class class class wbr 4018   omcom 4607   ~~ cen 6765   ~<_ cdom 6766   Fincfn 6767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-er 6560  df-en 6768  df-dom 6769  df-fin 6770
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator