Theorem clublem 37917

Description: If a superset Y of X possesses the property parameterized in x in ps, then Y is a superset of the closure of that property for the set X. (Contributed by RP, 23-Jul-2020.)

Hypotheses

Ref	Expression
clublem.y	|- ( ph -> Y e. _V )
clublem.sub	|- ( x = Y -> ( ps <-> ch ) )
clublem.sup	|- ( ph -> X C_ Y )
clublem.maj	|- ( ph -> ch )

Assertion

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

Distinct variable groups:   ch, x   x, X   x, Y

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem clublem

Step	Hyp	Ref	Expression
1		clublem.sup	. . 3 |- ( ph -> X C_ Y )
2		clublem.maj	. . 3 |- ( ph -> ch )
3		clublem.y	. . . . 5 |- ( ph -> Y e. _V )
4	3	a1d 25	. . . 4 |- ( ph -> ( ( X C_ Y /\ ch ) -> Y e. _V ) )
5		clublem.sub	. . . . . 6 |- ( x = Y -> ( ps <-> ch ) )
6	5	cleq2lem 37914	. . . . 5 |- ( x = Y -> ( ( X C_ x /\ ps ) <-> ( X C_ Y /\ ch ) ) )
7	6	elab3g 3357	. . . 4 |- ( ( ( X C_ Y /\ ch ) -> Y e. _V ) -> ( Y e. { x | ( X C_ x /\ ps ) } <-> ( X C_ Y /\ ch ) ) )
8	4, 7	syl 17	. . 3 |- ( ph -> ( Y e. { x | ( X C_ x /\ ps ) } <-> ( X C_ Y /\ ch ) ) )
9	1, 2, 8	mpbir2and 957	. 2 |- ( ph -> Y e. { x | ( X C_ x /\ ps ) } )
10		intss1 4492	. 2 |- ( Y e. { x | ( X C_ x /\ ps ) } -> |^| { x | ( X C_ x /\ ps ) } C_ Y )
11	9, 10	syl 17	1 |- ( ph -> |^| { x | ( X C_ x /\ ps ) } C_ Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   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-v 3202  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by:  mptrcllem  37920  trclubgNEW  37925