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Theorem domfin4 7953
Description: A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
domfin4  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  B  e. FinIV )

Proof of Theorem domfin4
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 domeng 6892 . . 3  |-  ( A  e. FinIV  ->  ( B  ~<_  A  <->  E. x ( B  ~~  x  /\  x  C_  A
) ) )
21biimpa 470 . 2  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  E. x ( B 
~~  x  /\  x  C_  A ) )
3 ensym 6926 . . . . . 6  |-  ( B 
~~  x  ->  x  ~~  B )
43ad2antrl 708 . . . . 5  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  ~~  B )
5 ssfin4 7952 . . . . . 6  |-  ( ( A  e. FinIV  /\  x  C_  A
)  ->  x  e. FinIV )
65ad2ant2rl 729 . . . . 5  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  e. FinIV )
7 fin4en1 7951 . . . . 5  |-  ( x 
~~  B  ->  (
x  e. FinIV  ->  B  e. FinIV ) )
84, 6, 7sylc 56 . . . 4  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  B  e. FinIV )
98ex 423 . . 3  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  ( ( B 
~~  x  /\  x  C_  A )  ->  B  e. FinIV
) )
109exlimdv 1626 . 2  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  ( E. x
( B  ~~  x  /\  x  C_  A )  ->  B  e. FinIV ) )
112, 10mpd 14 1  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  B  e. FinIV )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    e. wcel 1696    C_ wss 3165   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877  FinIVcfin4 7922
This theorem is referenced by:  infpssALT  7955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-fin4 7929
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