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Theorem isinffi 7814
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7260 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
isinffi  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. f  f : B -1-1-> A )
Distinct variable groups:    A, f    B, f

Proof of Theorem isinffi
Dummy variables  c 
a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ficardom 7783 . . 3  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
2 isinf 7260 . . 3  |-  ( -.  A  e.  Fin  ->  A. a  e.  om  E. c ( c  C_  A  /\  c  ~~  a
) )
3 breq2 4159 . . . . . 6  |-  ( a  =  ( card `  B
)  ->  ( c  ~~  a  <->  c  ~~  ( card `  B ) ) )
43anbi2d 685 . . . . 5  |-  ( a  =  ( card `  B
)  ->  ( (
c  C_  A  /\  c  ~~  a )  <->  ( c  C_  A  /\  c  ~~  ( card `  B )
) ) )
54exbidv 1633 . . . 4  |-  ( a  =  ( card `  B
)  ->  ( E. c ( c  C_  A  /\  c  ~~  a
)  <->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) ) )
65rspcva 2995 . . 3  |-  ( ( ( card `  B
)  e.  om  /\  A. a  e.  om  E. c ( c  C_  A  /\  c  ~~  a
) )  ->  E. c
( c  C_  A  /\  c  ~~  ( card `  B ) ) )
71, 2, 6syl2anr 465 . 2  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) )
8 simprr 734 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  ( card `  B ) )
9 ficardid 7784 . . . . . . 7  |-  ( B  e.  Fin  ->  ( card `  B )  ~~  B )
109ad2antlr 708 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( card `  B )  ~~  B )
11 entr 7097 . . . . . 6  |-  ( ( c  ~~  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  c  ~~  B )
128, 10, 11syl2anc 643 . . . . 5  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  B )
1312ensymd 7096 . . . 4  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  B  ~~  c )
14 bren 7055 . . . 4  |-  ( B 
~~  c  <->  E. f 
f : B -1-1-onto-> c )
1513, 14sylib 189 . . 3  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  E. f  f : B
-1-1-onto-> c )
16 f1of1 5615 . . . . . . 7  |-  ( f : B -1-1-onto-> c  ->  f : B -1-1-> c )
1716adantl 453 . . . . . 6  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  f : B -1-1-> c )
18 simplrl 737 . . . . . 6  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  c  C_  A )
19 f1ss 5586 . . . . . 6  |-  ( ( f : B -1-1-> c  /\  c  C_  A
)  ->  f : B -1-1-> A )
2017, 18, 19syl2anc 643 . . . . 5  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  f : B -1-1-> A )
2120ex 424 . . . 4  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( f : B -1-1-onto-> c  ->  f : B -1-1-> A
) )
2221eximdv 1629 . . 3  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( E. f  f : B -1-1-onto-> c  ->  E. f 
f : B -1-1-> A
) )
2315, 22mpd 15 . 2  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  E. f  f : B -1-1-> A )
247, 23exlimddv 1645 1  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. f  f : B -1-1-> A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2651    C_ wss 3265   class class class wbr 4155   omcom 4787   -1-1->wf1 5393   -1-1-onto->wf1o 5395   ` cfv 5396    ~~ cen 7044   Fincfn 7047   cardccrd 7757
This theorem is referenced by:  fidomtri  7815  hashdom  11582  erdsze2lem1  24670  eldioph2lem2  26512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761
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