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Theorem domfin4 7937
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 6876 . . 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 6910 . . . . . 6  |-  ( B 
~~  x  ->  x  ~~  B )
43ad2antrl 708 . . . . 5  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  ~~  B )
5 ssfin4 7936 . . . . . 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 7935 . . . . 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 1664 . 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 1528    e. wcel 1684    C_ wss 3152   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861  FinIVcfin4 7906
This theorem is referenced by:  infpssALT  7939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-fin4 7913
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