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Theorem domfi 7539
Description: A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
domfi  |-  ( ( A  e.  Fin  /\  B  ~<_  A )  ->  B  e.  Fin )

Proof of Theorem domfi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 domeng 7329 . . 3  |-  ( A  e.  Fin  ->  ( B  ~<_  A  <->  E. x
( B  ~~  x  /\  x  C_  A ) ) )
2 ssfi 7538 . . . . . . 7  |-  ( ( A  e.  Fin  /\  x  C_  A )  ->  x  e.  Fin )
32adantrl 715 . . . . . 6  |-  ( ( A  e.  Fin  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  x  e.  Fin )
4 enfii 7535 . . . . . . 7  |-  ( ( x  e.  Fin  /\  B  ~~  x )  ->  B  e.  Fin )
54adantrr 716 . . . . . 6  |-  ( ( x  e.  Fin  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  B  e.  Fin )
63, 5sylancom 667 . . . . 5  |-  ( ( A  e.  Fin  /\  ( B  ~~  x  /\  x  C_  A ) )  ->  B  e.  Fin )
76ex 434 . . . 4  |-  ( A  e.  Fin  ->  (
( B  ~~  x  /\  x  C_  A )  ->  B  e.  Fin ) )
87exlimdv 1690 . . 3  |-  ( A  e.  Fin  ->  ( E. x ( B  ~~  x  /\  x  C_  A
)  ->  B  e.  Fin ) )
91, 8sylbid 215 . 2  |-  ( A  e.  Fin  ->  ( B  ~<_  A  ->  B  e.  Fin ) )
109imp 429 1  |-  ( ( A  e.  Fin  /\  B  ~<_  A )  ->  B  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1586    e. wcel 1756    C_ wss 3333   class class class wbr 4297    ~~ cen 7312    ~<_ cdom 7313   Fincfn 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-om 6482  df-er 7106  df-en 7316  df-dom 7317  df-fin 7319
This theorem is referenced by:  xpfir  7540  dmfi  7599  fofi  7602  pwfilem  7610  pwfi  7611  sdom2en01  8476  isfin1-2  8559  fin67  8569  fin1a2lem9  8582  gchcda1  8828  hashdomi  12148  symggen  15981  cmpsub  19008  ufinffr  19507  alexsubALT  19628  ovolicc2lem4  21008  aannenlem1  21799  ffsrn  26034  harinf  29388  kelac2lem  29422
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