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Theorem fientri3 6381
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 6264 . . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
21biimpi 118 . . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
32adantr 270 . 2 |- ( ( A e. Fin /\ B e. Fin ) -> E. n e. om A ~~ n )
4 isfi 6264 . . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
54biimpi 118 . . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
65ad2antlr 472 . . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7 simplrr 502 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ n )
87adantr 270 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~~ n )
9 simpr 108 . . . . . . . 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 501 . . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n e. om )
1110adantr 270 . . . . . . . . 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 501 . . . . . . . . 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 6350 . . . . . . . . 9 |- ( ( n e. om /\ m e. om ) -> ( n ~<_ m <-> n C_ m ) )
1411, 12, 13syl2anc 403 . . . . . . . 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 165 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> n ~<_ m )
16 endomtr 6293 . . . . . . 7 |- ( ( A ~~ n /\ n ~<_ m ) -> A ~<_ m )
178, 15, 16syl2anc 403 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~<_ m )
18 simplrr 502 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> B ~~ m )
1918ensymd 6286 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> m ~~ B )
20 domentr 6294 . . . . . 6 |- ( ( A ~<_ m /\ m ~~ B ) -> A ~<_ B )
2117, 19, 20syl2anc 403 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> A ~<_ B )
2221orcd 684 . . . 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 502 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> B ~~ m )
24 simpr 108 . . . . . . . 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 501 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> m e. om )
2610adantr 270 . . . . . . . . 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 6350 . . . . . . . . 9 |- ( ( m e. om /\ n e. om ) -> ( m ~<_ n <-> m C_ n ) )
2825, 26, 27syl2anc 403 . . . . . . . 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 165 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> m ~<_ n )
30 endomtr 6293 . . . . . . 7 |- ( ( B ~~ m /\ m ~<_ n ) -> B ~<_ n )
3123, 29, 30syl2anc 403 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> B ~<_ n )
327adantr 270 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> A ~~ n )
3332ensymd 6286 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> n ~~ A )
34 domentr 6294 . . . . . 6 |- ( ( B ~<_ n /\ n ~~ A ) -> B ~<_ A )
3531, 33, 34syl2anc 403 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> B ~<_ A )
3635olcd 685 . . . 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 497 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
38 nntri2or2 6099 . . . . 5 |- ( ( n e. om /\ m e. om ) -> ( n C_ m \/ m C_ n ) )
3910, 37, 38syl2anc 403 . . . 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 744 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ B \/ B ~<_ A ) )
416, 40rexlimddv 2481 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( A ~<_ B \/ B ~<_ A ) )
423, 41rexlimddv 2481 1 |- ( ( A e. Fin /\ B e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   e. wcel 1433   E.wrex 2349   C_ wss 2973   class class class wbr 3785   omcom 4331   ~~ cen 6242   ~<_ cdom 6243   Fincfn 6244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329
This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-er 6129  df-en 6245  df-dom 6246  df-fin 6247
This theorem is referenced by: (None)
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