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Theorem isinffi 8404
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7767 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 8373 . . 3  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
2 isinf 7767 . . 3  |-  ( -.  A  e.  Fin  ->  A. a  e.  om  E. c ( c  C_  A  /\  c  ~~  a
) )
3 breq2 4398 . . . . . 6  |-  ( a  =  ( card `  B
)  ->  ( c  ~~  a  <->  c  ~~  ( card `  B ) ) )
43anbi2d 702 . . . . 5  |-  ( a  =  ( card `  B
)  ->  ( (
c  C_  A  /\  c  ~~  a )  <->  ( c  C_  A  /\  c  ~~  ( card `  B )
) ) )
54exbidv 1735 . . . 4  |-  ( a  =  ( card `  B
)  ->  ( E. c ( c  C_  A  /\  c  ~~  a
)  <->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) ) )
65rspcva 3157 . . 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 476 . 2  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) )
8 simprr 758 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  ( card `  B ) )
9 ficardid 8374 . . . . . . 7  |-  ( B  e.  Fin  ->  ( card `  B )  ~~  B )
109ad2antlr 725 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( card `  B )  ~~  B )
11 entr 7604 . . . . . 6  |-  ( ( c  ~~  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  c  ~~  B )
128, 10, 11syl2anc 659 . . . . 5  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  B )
1312ensymd 7603 . . . 4  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  B  ~~  c )
14 bren 7562 . . . 4  |-  ( B 
~~  c  <->  E. f 
f : B -1-1-onto-> c )
1513, 14sylib 196 . . 3  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  E. f  f : B
-1-1-onto-> c )
16 f1of1 5797 . . . . . . 7  |-  ( f : B -1-1-onto-> c  ->  f : B -1-1-> c )
1716adantl 464 . . . . . 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 762 . . . . . 6  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  c  C_  A )
19 f1ss 5768 . . . . . 6  |-  ( ( f : B -1-1-> c  /\  c  C_  A
)  ->  f : B -1-1-> A )
2017, 18, 19syl2anc 659 . . . . 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 432 . . . 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 1731 . . 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 1747 1  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. f  f : B -1-1-> A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   A.wral 2753    C_ wss 3413   class class class wbr 4394   -1-1->wf1 5565   -1-1-onto->wf1o 5567   ` cfv 5568   omcom 6682    ~~ cen 7550   Fincfn 7553   cardccrd 8347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351
This theorem is referenced by:  fidomtri  8405  hashdom  12493  erdsze2lem1  29487  eldioph2lem2  35035
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