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Theorem fientri3 6880
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 6727 . . . 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 6727 . . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
54biimpi 119 . . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
65ad2antlr 481 . . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7 simplrr 526 . . . . . . . 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 525 . . . . . . . . . 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 525 . . . . . . . . 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 6830 . . . . . . . . 9 |- ( ( n e. om /\ m e. om ) -> ( n ~<_ m <-> n C_ m ) )
1411, 12, 13syl2anc 409 . . . . . . . 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 6756 . . . . . . 7 |- ( ( A ~~ n /\ n ~<_ m ) -> A ~<_ m )
178, 15, 16syl2anc 409 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~<_ m )
18 simplrr 526 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> B ~~ m )
1918ensymd 6749 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> m ~~ B )
20 domentr 6757 . . . . . 6 |- ( ( A ~<_ m /\ m ~~ B ) -> A ~<_ B )
2117, 19, 20syl2anc 409 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~<_ B )
2221orcd 723 . . . 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 526 . . . . . . 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 525 . . . . . . . . 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 6830 . . . . . . . . 9 |- ( ( m e. om /\ n e. om ) -> ( m ~<_ n <-> m C_ n ) )
2825, 26, 27syl2anc 409 . . . . . . . 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 6756 . . . . . . 7 |- ( ( B ~~ m /\ m ~<_ n ) -> B ~<_ n )
3123, 29, 30syl2anc 409 . . . . . 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 6749 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> n ~~ A )
34 domentr 6757 . . . . . 6 |- ( ( B ~<_ n /\ n ~~ A ) -> B ~<_ A )
3531, 33, 34syl2anc 409 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> B ~<_ A )
3635olcd 724 . . . 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 521 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
38 nntri2or2 6466 . . . . 5 |- ( ( n e. om /\ m e. om ) -> ( n C_ m \/ m C_ n ) )
3910, 37, 38syl2anc 409 . . . 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 788 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ B \/ B ~<_ A ) )
416, 40rexlimddv 2588 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( A ~<_ B \/ B ~<_ A ) )
423, 41rexlimddv 2588 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 698   e. wcel 2136   E.wrex 2445   C_ wss 3116   class class class wbr 3982   omcom 4567   ~~ cen 6704   ~<_ cdom 6705   Fincfn 6706
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709
This theorem is referenced by: (None)
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