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Theorem ttukeylem2 8137
Description: Lemma for ttukey 8145. 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 447 . . . . . 6  |-  ( (
ph  /\  D  C_  C
)  ->  D  C_  C
)
2 sspwb 4223 . . . . . 6  |-  ( D 
C_  C  <->  ~P D  C_ 
~P C )
31, 2sylib 188 . . . . 5  |-  ( (
ph  /\  D  C_  C
)  ->  ~P D  C_ 
~P C )
4 ssrin 3394 . . . . 5  |-  ( ~P D  C_  ~P C  ->  ( ~P D  i^i  Fin )  C_  ( ~P C  i^i  Fin ) )
5 sstr2 3186 . . . . 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 18 . . . 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 8136 . . . . 5  |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
1110adantr 451 . . . 4  |-  ( (
ph  /\  D  C_  C
)  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A
) )
127, 8, 9ttukeylem1 8136 . . . . 5  |-  ( ph  ->  ( D  e.  A  <->  ( ~P D  i^i  Fin )  C_  A ) )
1312adantr 451 . . . 4  |-  ( (
ph  /\  D  C_  C
)  ->  ( D  e.  A  <->  ( ~P D  i^i  Fin )  C_  A
) )
146, 11, 133imtr4d 259 . . 3  |-  ( (
ph  /\  D  C_  C
)  ->  ( C  e.  A  ->  D  e.  A ) )
1514impancom 427 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( D  C_  C  ->  D  e.  A ) )
1615impr 602 1  |-  ( (
ph  /\  ( C  e.  A  /\  D  C_  C ) )  ->  D  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    e. wcel 1684    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   -1-1-onto->wf1o 5254   ` cfv 5255   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  ttukeylem6  8141  ttukeylem7  8142
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-3or 935  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-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-1o 6479  df-en 6864  df-dom 6865  df-fin 6867
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