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Theorem fientri3 6916
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 6763 . . . 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 6763 . . . . 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 6866 . . . . . . . . 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 6792 . . . . . . 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 6785 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> m ~~ B )
20 domentr 6793 . . . . . 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 733 . . . 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 6866 . . . . . . . . 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 6792 . . . . . . 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 6785 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> n ~~ A )
34 domentr 6793 . . . . . 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 734 . . . 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 6501 . . . . 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 798 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ B \/ B ~<_ A ) )
416, 40rexlimddv 2599 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( A ~<_ B \/ B ~<_ A ) )
423, 41rexlimddv 2599 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 708   e. wcel 2148   E.wrex 2456   C_ wss 3131   class class class wbr 4005   omcom 4591   ~~ cen 6740   ~<_ cdom 6741   Fincfn 6742
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745
This theorem is referenced by: (None)
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