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Theorem domfin4 8643
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 7488 . . 3  |-  ( A  e. FinIV  ->  ( B  ~<_  A  <->  E. x ( B  ~~  x  /\  x  C_  A
) ) )
21biimpa 482 . 2  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  E. x ( B 
~~  x  /\  x  C_  A ) )
3 ensym 7522 . . . 4  |-  ( B 
~~  x  ->  x  ~~  B )
43ad2antrl 726 . . 3  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  ~~  B )
5 ssfin4 8642 . . . 4  |-  ( ( A  e. FinIV  /\  x  C_  A
)  ->  x  e. FinIV )
65ad2ant2rl 747 . . 3  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  e. FinIV )
7 fin4en1 8641 . . 3  |-  ( x 
~~  B  ->  (
x  e. FinIV  ->  B  e. FinIV ) )
84, 6, 7sylc 59 . 2  |-  ( ( ( A  e. FinIV  /\  B  ~<_  A )  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  B  e. FinIV )
92, 8exlimddv 1747 1  |-  ( ( A  e. FinIV  /\  B  ~<_  A )  ->  B  e. FinIV )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1633    e. wcel 1842    C_ wss 3413   class class class wbr 4394    ~~ cen 7471    ~<_ cdom 7472  FinIVcfin4 8612
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-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  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-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7268  df-en 7475  df-dom 7476  df-fin4 8619
This theorem is referenced by:  infpssALT  8645
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