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Theorem isinffi 7871
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7314 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 7840 . . 3  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
2 isinf 7314 . . 3  |-  ( -.  A  e.  Fin  ->  A. a  e.  om  E. c ( c  C_  A  /\  c  ~~  a
) )
3 breq2 4208 . . . . . 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 1636 . . . 4  |-  ( a  =  ( card `  B
)  ->  ( E. c ( c  C_  A  /\  c  ~~  a
)  <->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) ) )
65rspcva 3042 . . 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 7841 . . . . . . 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 7151 . . . . . 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 7150 . . . 4  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  B  ~~  c )
14 bren 7109 . . . 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 5665 . . . . . . 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 5636 . . . . . 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 1632 . . 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 1648 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 1550    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   omcom 4837   -1-1->wf1 5443   -1-1-onto->wf1o 5445   ` cfv 5446    ~~ cen 7098   Fincfn 7101   cardccrd 7814
This theorem is referenced by:  fidomtri  7872  hashdom  11645  erdsze2lem1  24881  eldioph2lem2  26800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818
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