Theorem disjss1 4070

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 3221	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 336	. . . . 5 |- ( A C_ B -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) )
3	2	alrimiv 1922	. . . 4 |- ( A C_ B -> A. x ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) )
4		moim 2144	. . . 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 14	. . 3 |- ( A C_ B -> ( E* x ( x e. B /\ y e. C ) -> E* x ( x e. A /\ y e. C ) ) )
6	5	alimdv 1927	. 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 4066	. 2 |- (Disj x e. B C <-> A. y E* x ( x e. B /\ y e. C ) )
8		dfdisj2 4066	. 2 |- (Disj x e. A C <-> A. y E* x ( x e. A /\ y e. C ) )
9	6, 7, 8	3imtr4g 205	1 |- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1395   E*wmo 2080   e. wcel 2202   C_ wss 3200  Disj wdisj 4064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-rmo 2518  df-in 3206  df-ss 3213  df-disj 4065

This theorem is referenced by:  disjeq1  4071  disjx0  4087  fsumiun  12037