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Theorem fientri3 6892
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 6739 . . . 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 6739 . . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
54biimpi 119 . . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
65ad2antlr 486 . . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7 simplrr 531 . . . . . . . 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 530 . . . . . . . . . 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 530 . . . . . . . . 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 6842 . . . . . . . . 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 6768 . . . . . . 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 531 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> B ~~ m )
1918ensymd 6761 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ n C_ m ) -> m ~~ B )
20 domentr 6769 . . . . . 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 728 . . . 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 531 . . . . . . 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 530 . . . . . . . . 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 6842 . . . . . . . . 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 6768 . . . . . . 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 6761 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) /\ m C_ n ) -> n ~~ A )
34 domentr 6769 . . . . . 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 729 . . . 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 526 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
38 nntri2or2 6477 . . . . 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 793 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ B \/ B ~<_ A ) )
416, 40rexlimddv 2592 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( A ~<_ B \/ B ~<_ A ) )
423, 41rexlimddv 2592 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 703   e. wcel 2141   E.wrex 2449   C_ wss 3121   class class class wbr 3989   omcom 4574   ~~ cen 6716   ~<_ cdom 6717   Fincfn 6718
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721
This theorem is referenced by: (None)
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