Theorem bj-ismoored2 33063

Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)

Hypotheses

Ref	Expression
bj-ismoored.1	|- ( ph -> A e. Moore_ )
bj-ismoored.2	|- ( ph -> B C_ A )
bj-ismoored2.3	|- ( ph -> B =/= (/) )

Assertion

Ref	Expression
bj-ismoored2	|- ( ph -> |^| B e. A )

Proof of Theorem bj-ismoored2

Step	Hyp	Ref	Expression
1		bj-ismoored.2	. . . 4 |- ( ph -> B C_ A )
2		bj-ismoored2.3	. . . 4 |- ( ph -> B =/= (/) )
3		intssuni2 4502	. . . 4 |- ( ( B C_ A /\ B =/= (/) ) -> |^| B C_ U. A )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> |^| B C_ U. A )
5		sseqin2 3817	. . 3 |- ( |^| B C_ U. A <-> ( U. A i^i |^| B ) = |^| B )
6	4, 5	sylib 208	. 2 |- ( ph -> ( U. A i^i |^| B ) = |^| B )
7		bj-ismoored.1	. . 3 |- ( ph -> A e. Moore_ )
8	7, 1	bj-ismoored 33062	. 2 |- ( ph -> ( U. A i^i |^| B ) e. A )
9	6, 8	eqeltrrd 2702	1 |- ( ph -> |^| B e. A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475  Moore_cmoore 33057

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  ax-sep 4781

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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-int 4476  df-bj-moore 33058

This theorem is referenced by: (None)