Theorem elpg 42457

Description: Membership in the class of partizan games. In ONAG this is stated as "If L and R are any two sets of games, then there is a game { L | R }. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.)

Assertion

Ref	Expression
elpg	|- ( A e. Pg <-> ( A e. ( _V X. _V ) /\ ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )

Proof of Theorem elpg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpglem1 42454	. . . 4 |- ( E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) -> ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )
2		elpglem2 42455	. . . 4 |- ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) -> E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) )
3	1, 2	impbii 199	. . 3 |- ( E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) <-> ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )
4	3	anbi2i 730	. 2 |- ( ( A e. ( _V X. _V ) /\ E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) ) )
5		df-pg 42453	. . . 4 |- Pg = setrecs ( ( y e. _V |-> ( ~P y X. ~P y ) ) )
6	5	elsetrecs 42445	. . 3 |- ( A e. Pg <-> E. x ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) )
7		elpglem3 42456	. . 3 |- ( E. x ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) <-> ( A e. ( _V X. _V ) /\ E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
8	6, 7	bitri 264	. 2 |- ( A e. Pg <-> ( A e. ( _V X. _V ) /\ E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
9		3anass 1042	. 2 |- ( ( A e. ( _V X. _V ) /\ ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) ) )
10	4, 8, 9	3bitr4i 292	1 |- ( A e. Pg <-> ( A e. ( _V X. _V ) /\ ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  Pgcpg 42452

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  ax-reg 8497  ax-inf2 8538

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-int 4476  df-iun 4522  df-iin 4523  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-pred 5680  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628  df-setrecs 42431  df-pg 42453

This theorem is referenced by: (None)