Theorem disjss1f 29386

Description: A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Hypotheses

Ref	Expression
disjss1f.1	|- F/_ x A
disjss1f.2	|- F/_ x B

Assertion

Ref	Expression
disjss1f	|- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )

Proof of Theorem disjss1f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjss1f.1	. . . 4 |- F/_ x A
2		disjss1f.2	. . . 4 |- F/_ x B
3	1, 2	ssrmo 29334	. . 3 |- ( A C_ B -> ( E* x e. B y e. C -> E* x e. A y e. C ) )
4	3	alimdv 1845	. 2 |- ( A C_ B -> ( A. y E* x e. B y e. C -> A. y E* x e. A y e. C ) )
5		df-disj 4621	. 2 |- (Disj x e. B C <-> A. y E* x e. B y e. C )
6		df-disj 4621	. 2 |- (Disj x e. A C <-> A. y E* x e. A y e. C )
7	4, 5, 6	3imtr4g 285	1 |- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   F/_wnfc 2751   E*wrmo 2915   C_ wss 3574  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-in 3581  df-ss 3588  df-disj 4621

This theorem is referenced by:  disjeq1f  29387  esumrnmpt2  30130  measvuni  30277