Theorem elpglem1 42454

Description: Lemma for elpg 42457. (Contributed by Emmett Weisz, 28-Aug-2021.)

Assertion

Ref	Expression
elpglem1	|- ( E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) -> ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )

Distinct variable group:   x, A

Proof of Theorem elpglem1

Step	Hyp	Ref	Expression
1		elpwi 4168	. . . . 5 |- ( ( 1st ` A ) e. ~P x -> ( 1st ` A ) C_ x )
2	1	adantl 482	. . . 4 |- ( ( x C_ Pg /\ ( 1st ` A ) e. ~P x ) -> ( 1st ` A ) C_ x )
3		simpl 473	. . . 4 |- ( ( x C_ Pg /\ ( 1st ` A ) e. ~P x ) -> x C_ Pg )
4	2, 3	sstrd 3613	. . 3 |- ( ( x C_ Pg /\ ( 1st ` A ) e. ~P x ) -> ( 1st ` A ) C_ Pg )
5		elpwi 4168	. . . . 5 |- ( ( 2nd ` A ) e. ~P x -> ( 2nd ` A ) C_ x )
6	5	adantl 482	. . . 4 |- ( ( x C_ Pg /\ ( 2nd ` A ) e. ~P x ) -> ( 2nd ` A ) C_ x )
7		simpl 473	. . . 4 |- ( ( x C_ Pg /\ ( 2nd ` A ) e. ~P x ) -> x C_ Pg )
8	6, 7	sstrd 3613	. . 3 |- ( ( x C_ Pg /\ ( 2nd ` A ) e. ~P x ) -> ( 2nd ` A ) C_ Pg )
9	4, 8	anim12dan 882	. 2 |- ( ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) -> ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )
10	9	exlimiv 1858	1 |- ( E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) -> ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   ` 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  elpg  42457