Theorem ttukeylem2 9332

Description: Lemma for ttukey 9340. 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

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( ph /\ D C_ C ) -> D C_ C )
2		sspwb 4917	. . . . . 6 |- ( D C_ C <-> ~P D C_ ~P C )
3	1, 2	sylib 208	. . . . 5 |- ( ( ph /\ D C_ C ) -> ~P D C_ ~P C )
4		ssrin 3838	. . . . 5 |- ( ~P D C_ ~P C -> ( ~P D i^i Fin ) C_ ( ~P C i^i Fin ) )
5		sstr2 3610	. . . . 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 ) )
6	3, 4, 5	3syl 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 ) )
10	7, 8, 9	ttukeylem1 9331	. . . . 5 |- ( ph -> ( C e. A <-> ( ~P C i^i Fin ) C_ A ) )
11	10	adantr 481	. . . 4 |- ( ( ph /\ D C_ C ) -> ( C e. A <-> ( ~P C i^i Fin ) C_ A ) )
12	7, 8, 9	ttukeylem1 9331	. . . . 5 |- ( ph -> ( D e. A <-> ( ~P D i^i Fin ) C_ A ) )
13	12	adantr 481	. . . 4 |- ( ( ph /\ D C_ C ) -> ( D e. A <-> ( ~P D i^i Fin ) C_ A ) )
14	6, 11, 13	3imtr4d 283	. . 3 |- ( ( ph /\ D C_ C ) -> ( C e. A -> D e. A ) )
15	14	impancom 456	. 2 |- ( ( ph /\ C e. A ) -> ( D C_ C -> D e. A ) )
16	15	impr 649	1 |- ( ( ph /\ ( C e. A /\ D C_ C ) ) -> D e. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   -1-1-onto->wf1o 5887   ` cfv 5888   Fincfn 7955   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-en 7956  df-dom 7957  df-fin 7959

This theorem is referenced by:  ttukeylem6  9336  ttukeylem7  9337