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Theorem domfin4 8687
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 7527 . . 3  |-  ( A  e. FinIV  ->  ( B  ~<_  A  <->  E. x ( B  ~~  x  /\  x  C_  A
) ) )
21biimpa 484 . 2  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  E. x ( B 
~~  x  /\  x  C_  A ) )
3 ensym 7561 . . . 4  |-  ( B 
~~  x  ->  x  ~~  B )
43ad2antrl 727 . . 3  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  ~~  B )
5 ssfin4 8686 . . . 4  |-  ( ( A  e. FinIV  /\  x  C_  A
)  ->  x  e. FinIV )
65ad2ant2rl 748 . . 3  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  e. FinIV )
7 fin4en1 8685 . . 3  |-  ( x 
~~  B  ->  (
x  e. FinIV  ->  B  e. FinIV ) )
84, 6, 7sylc 60 . 2  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  B  e. FinIV )
92, 8exlimddv 1702 1  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  B  e. FinIV )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1596    e. wcel 1767    C_ wss 3476   class class class wbr 4447    ~~ cen 7510    ~<_ cdom 7511  FinIVcfin4 8656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-fin4 8663
This theorem is referenced by:  infpssALT  8689
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