Theorem disjin2 29400

Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Assertion

Ref	Expression
disjin2	|- (Disj x e. B C -> Disj x e. B ( A i^i C ) )

Proof of Theorem disjin2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elinel2 3800	. . . . . 6 |- ( y e. ( A i^i C ) -> y e. C )
2	1	anim2i 593	. . . . 5 |- ( ( x e. B /\ y e. ( A i^i C ) ) -> ( x e. B /\ y e. C ) )
3	2	ax-gen 1722	. . . 4 |- A. x ( ( x e. B /\ y e. ( A i^i C ) ) -> ( x e. B /\ y e. C ) )
4	3	rmoimi2 3409	. . 3 |- ( E* x e. B y e. C -> E* x e. B y e. ( A i^i C ) )
5	4	alimi 1739	. 2 |- ( A. y E* x e. B y e. C -> A. y E* x e. B y e. ( A i^i C ) )
6		df-disj 4621	. 2 |- (Disj x e. B C <-> A. y E* x e. B y e. C )
7		df-disj 4621	. 2 |- (Disj x e. B ( A i^i C ) <-> A. y E* x e. B y e. ( A i^i C ) )
8	5, 6, 7	3imtr4i 281	1 |- (Disj x e. B C -> Disj x e. B ( A i^i C ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   E*wrmo 2915   i^i cin 3573  Disj wdisj 4620

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920  df-v 3202  df-in 3581  df-disj 4621

This theorem is referenced by:  ldgenpisyslem1  30226