Theorem clss2lem 37918

Description: The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)

Hypothesis

Ref	Expression
clss2lem.1	|- ( ph -> ( ch -> ps ) )

Assertion

Ref	Expression
clss2lem	|- ( ph -> |^| { x | ( X C_ x /\ ps ) } C_ |^| { x | ( X C_ x /\ ch ) } )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   X( x)

Proof of Theorem clss2lem

Step	Hyp	Ref	Expression
1		clss2lem.1	. . . . 5 |- ( ph -> ( ch -> ps ) )
2	1	adantld 483	. . . 4 |- ( ph -> ( ( X C_ x /\ ch ) -> ps ) )
3	2	alrimiv 1855	. . 3 |- ( ph -> A. x ( ( X C_ x /\ ch ) -> ps ) )
4		pm5.3 748	. . . . 5 |- ( ( ( X C_ x /\ ch ) -> ps ) <-> ( ( X C_ x /\ ch ) -> ( X C_ x /\ ps ) ) )
5	4	albii 1747	. . . 4 |- ( A. x ( ( X C_ x /\ ch ) -> ps ) <-> A. x ( ( X C_ x /\ ch ) -> ( X C_ x /\ ps ) ) )
6		ss2ab 3670	. . . 4 |- ( { x | ( X C_ x /\ ch ) } C_ { x | ( X C_ x /\ ps ) } <-> A. x ( ( X C_ x /\ ch ) -> ( X C_ x /\ ps ) ) )
7	5, 6	bitr4i 267	. . 3 |- ( A. x ( ( X C_ x /\ ch ) -> ps ) <-> { x | ( X C_ x /\ ch ) } C_ { x | ( X C_ x /\ ps ) } )
8	3, 7	sylib 208	. 2 |- ( ph -> { x | ( X C_ x /\ ch ) } C_ { x | ( X C_ x /\ ps ) } )
9		intss 4498	. 2 |- ( { x | ( X C_ x /\ ch ) } C_ { x | ( X C_ x /\ ps ) } -> |^| { x | ( X C_ x /\ ps ) } C_ |^| { x | ( X C_ x /\ ch ) } )
10	8, 9	syl 17	1 |- ( ph -> |^| { x | ( X C_ x /\ ps ) } C_ |^| { x | ( X C_ x /\ ch ) } )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   {cab 2608   C_ wss 3574   |^|cint 4475

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-ral 2917  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by: (None)