Theorem disjss1 4413

Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjss1	|- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem disjss1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3483	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 564	. . . . 5 |- ( A C_ B -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) )
3	2	alrimiv 1706	. . . 4 |- ( A C_ B -> A. x ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) )
4		moim 2325	. . . 4 |- ( A. x ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) -> ( E* x ( x e. B /\ y e. C ) -> E* x ( x e. A /\ y e. C ) ) )
5	3, 4	syl 16	. . 3 |- ( A C_ B -> ( E* x ( x e. B /\ y e. C ) -> E* x ( x e. A /\ y e. C ) ) )
6	5	alimdv 1696	. 2 |- ( A C_ B -> ( A. y E* x ( x e. B /\ y e. C ) -> A. y E* x ( x e. A /\ y e. C ) ) )
7		dfdisj2 4409	. 2 |- (Disj x e. B C <-> A. y E* x ( x e. B /\ y e. C ) )
8		dfdisj2 4409	. 2 |- (Disj x e. A C <-> A. y E* x ( x e. A /\ y e. C ) )
9	6, 7, 8	3imtr4g 270	1 |- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1381   e. wcel 1804   E*wmo 2269   C_ wss 3461  Disj wdisj 4407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-rmo 2801  df-in 3468  df-ss 3475  df-disj 4408

This theorem is referenced by:  disjeq1  4414  disjx0  4432  disjxiun  4434  disjss3  4436  volfiniun  21935  uniioovol  21966  uniioombllem4  21973  sibfof  28260