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Theorem ttukeylem2 8323
Description: Lemma for ttukey 8331. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1  |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )
ttukeylem.2  |-  ( ph  ->  B  e.  A )
ttukeylem.3  |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )
Assertion
Ref Expression
ttukeylem2  |-  ( (
ph  /\  ( C  e.  A  /\  D  C_  C ) )  ->  D  e.  A )
Distinct variable groups:    x, C    x, D    x, A    x, B    x, F
Allowed substitution hint:    ph( x)

Proof of Theorem ttukeylem2
StepHypRef Expression
1 simpr 448 . . . . . 6  |-  ( (
ph  /\  D  C_  C
)  ->  D  C_  C
)
2 sspwb 4354 . . . . . 6  |-  ( D 
C_  C  <->  ~P D  C_ 
~P C )
31, 2sylib 189 . . . . 5  |-  ( (
ph  /\  D  C_  C
)  ->  ~P D  C_ 
~P C )
4 ssrin 3509 . . . . 5  |-  ( ~P D  C_  ~P C  ->  ( ~P D  i^i  Fin )  C_  ( ~P C  i^i  Fin ) )
5 sstr2 3298 . . . . 5  |-  ( ( ~P D  i^i  Fin )  C_  ( ~P C  i^i  Fin )  ->  (
( ~P C  i^i  Fin )  C_  A  ->  ( ~P D  i^i  Fin )  C_  A ) )
63, 4, 53syl 19 . . . 4  |-  ( (
ph  /\  D  C_  C
)  ->  ( ( ~P C  i^i  Fin )  C_  A  ->  ( ~P D  i^i  Fin )  C_  A ) )
7 ttukeylem.1 . . . . . 6  |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )
8 ttukeylem.2 . . . . . 6  |-  ( ph  ->  B  e.  A )
9 ttukeylem.3 . . . . . 6  |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )
107, 8, 9ttukeylem1 8322 . . . . 5  |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
1110adantr 452 . . . 4  |-  ( (
ph  /\  D  C_  C
)  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A
) )
127, 8, 9ttukeylem1 8322 . . . . 5  |-  ( ph  ->  ( D  e.  A  <->  ( ~P D  i^i  Fin )  C_  A ) )
1312adantr 452 . . . 4  |-  ( (
ph  /\  D  C_  C
)  ->  ( D  e.  A  <->  ( ~P D  i^i  Fin )  C_  A
) )
146, 11, 133imtr4d 260 . . 3  |-  ( (
ph  /\  D  C_  C
)  ->  ( C  e.  A  ->  D  e.  A ) )
1514impancom 428 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( D  C_  C  ->  D  e.  A ) )
1615impr 603 1  |-  ( (
ph  /\  ( C  e.  A  /\  D  C_  C ) )  ->  D  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    e. wcel 1717    \ cdif 3260    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   -1-1-onto->wf1o 5393   ` cfv 5394   Fincfn 7045   cardccrd 7755
This theorem is referenced by:  ttukeylem6  8327  ttukeylem7  8328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1o 6660  df-en 7046  df-dom 7047  df-fin 7049
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