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Theorem ttukeylem2 8139
Description: Lemma for ttukey 8147. 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 4225 . . . . . 6  |-  ( D 
C_  C  <->  ~P D  C_ 
~P C )
31, 2sylib 188 . . . . 5  |-  ( (
ph  /\  D  C_  C
)  ->  ~P D  C_ 
~P C )
4 ssrin 3396 . . . . 5  |-  ( ~P D  C_  ~P C  ->  ( ~P D  i^i  Fin )  C_  ( ~P C  i^i  Fin ) )
5 sstr2 3188 . . . . 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 8138 . . . . 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 8138 . . . . 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 1529    e. wcel 1686    \ cdif 3151    i^i cin 3153    C_ wss 3154   ~Pcpw 3627   U.cuni 3829   -1-1-onto->wf1o 5256   ` cfv 5257   Fincfn 6865   cardccrd 7570
This theorem is referenced by:  ttukeylem6  8143  ttukeylem7  8144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-1o 6481  df-en 6866  df-dom 6867  df-fin 6869
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