Theorem disjss1 4091

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 3232	. . . . . 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 1923	. . . 4 |- ( A C_ B -> A. x ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) )
4		moim 2145	. . . 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 1928	. 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 4087	. 2 |- (Disj x e. B C <-> A. y E* x ( x e. B /\ y e. C ) )
8		dfdisj2 4087	. 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 1396   E*wmo 2081   e. wcel 2203   C_ wss 3211  Disj wdisj 4085

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-rmo 2528  df-in 3217  df-ss 3224  df-disj 4086

This theorem is referenced by:  disjeq1  4092  disjx0  4108  fsumiun  12163