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Theorem fisbth 6939
Description: Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
Assertion
Ref Expression
fisbth |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
Proof of Theorem fisbth
Dummy variables m n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6815 . . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
21biimpi 120 . . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
32ad2antrr 488 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> E. n e. om A ~~ n )
4 isfi 6815 . . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
54biimpi 120 . . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
65ad3antlr 493 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7 simplrr 536 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ n )
87ensymd 6837 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
9 simprl 529 . . . . . . . . . 10 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~<_ B )
109ad2antrr 488 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~<_ B )
11 endomtr 6844 . . . . . . . . 9 |- ( ( n ~~ A /\ A ~<_ B ) -> n ~<_ B )
128, 10, 11syl2anc 411 . . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~<_ B )
13 simprr 531 . . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> B ~~ m )
14 domentr 6845 . . . . . . . 8 |- ( ( n ~<_ B /\ B ~~ m ) -> n ~<_ m )
1512, 13, 14syl2anc 411 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~<_ m )
16 simplrl 535 . . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n e. om )
17 simprl 529 . . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
18 nndomo 6920 . . . . . . . 8 |- ( ( n e. om /\ m e. om ) -> ( n ~<_ m <-> n C_ m ) )
1916, 17, 18syl2anc 411 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( n ~<_ m <-> n C_ m ) )
2015, 19mpbid 147 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n C_ m )
2113ensymd 6837 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
22 simprr 531 . . . . . . . . . 10 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> B ~<_ A )
2322ad2antrr 488 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> B ~<_ A )
24 endomtr 6844 . . . . . . . . 9 |- ( ( m ~~ B /\ B ~<_ A ) -> m ~<_ A )
2521, 23, 24syl2anc 411 . . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~<_ A )
26 domentr 6845 . . . . . . . 8 |- ( ( m ~<_ A /\ A ~~ n ) -> m ~<_ n )
2725, 7, 26syl2anc 411 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~<_ n )
28 nndomo 6920 . . . . . . . 8 |- ( ( m e. om /\ n e. om ) -> ( m ~<_ n <-> m C_ n ) )
2917, 16, 28syl2anc 411 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( m ~<_ n <-> m C_ n ) )
3027, 29mpbid 147 . . . . . 6 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m C_ n )
3120, 30eqssd 3196 . . . . 5 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n = m )
327, 31breqtrd 4055 . . . 4 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ m )
33 entr 6838 . . . 4 |- ( ( A ~~ m /\ m ~~ B ) -> A ~~ B )
3432, 21, 33syl2anc 411 . . 3 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ B )
356, 34rexlimddv 2616 . 2 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) /\ ( n e. om /\ A ~~ n ) ) -> A ~~ B )
363, 35rexlimddv 2616 1 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   E.wrex 2473   C_ wss 3153   class class class wbr 4029   omcom 4622   ~~ cen 6792   ~<_ cdom 6793   Fincfn 6794
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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797
This theorem is referenced by: (None)
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