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Theorem isinffi 8266
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7630 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 8235 . . 3  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
2 isinf 7630 . . 3  |-  ( -.  A  e.  Fin  ->  A. a  e.  om  E. c ( c  C_  A  /\  c  ~~  a
) )
3 breq2 4397 . . . . . 6  |-  ( a  =  ( card `  B
)  ->  ( c  ~~  a  <->  c  ~~  ( card `  B ) ) )
43anbi2d 703 . . . . 5  |-  ( a  =  ( card `  B
)  ->  ( (
c  C_  A  /\  c  ~~  a )  <->  ( c  C_  A  /\  c  ~~  ( card `  B )
) ) )
54exbidv 1681 . . . 4  |-  ( a  =  ( card `  B
)  ->  ( E. c ( c  C_  A  /\  c  ~~  a
)  <->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) ) )
65rspcva 3170 . . 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 478 . 2  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) )
8 simprr 756 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  ( card `  B ) )
9 ficardid 8236 . . . . . . 7  |-  ( B  e.  Fin  ->  ( card `  B )  ~~  B )
109ad2antlr 726 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( card `  B )  ~~  B )
11 entr 7464 . . . . . 6  |-  ( ( c  ~~  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  c  ~~  B )
128, 10, 11syl2anc 661 . . . . 5  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  B )
1312ensymd 7463 . . . 4  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  B  ~~  c )
14 bren 7422 . . . 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 5741 . . . . . . 7  |-  ( f : B -1-1-onto-> c  ->  f : B -1-1-> c )
1716adantl 466 . . . . . 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 759 . . . . . 6  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  c  C_  A )
19 f1ss 5712 . . . . . 6  |-  ( ( f : B -1-1-> c  /\  c  C_  A
)  ->  f : B -1-1-> A )
2017, 18, 19syl2anc 661 . . . . 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 434 . . . 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 1677 . . 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 1693 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 369    = wceq 1370   E.wex 1587    e. wcel 1758   A.wral 2795    C_ wss 3429   class class class wbr 4393   -1-1->wf1 5516   -1-1-onto->wf1o 5518   ` cfv 5519   omcom 6579    ~~ cen 7410   Fincfn 7413   cardccrd 8209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-om 6580  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213
This theorem is referenced by:  fidomtri  8267  hashdom  12253  erdsze2lem1  27228  eldioph2lem2  29240
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