Theorem disjss1 4371

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 3435	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 562	. . . . 5 |- ( A C_ B -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) )
3	2	alrimiv 1740	. . . 4 |- ( A C_ B -> A. x ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. C ) ) )
4		moim 2291	. . . 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 17	. . 3 |- ( A C_ B -> ( E* x ( x e. B /\ y e. C ) -> E* x ( x e. A /\ y e. C ) ) )
6	5	alimdv 1730	. 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 4367	. 2 |- (Disj x e. B C <-> A. y E* x ( x e. B /\ y e. C ) )
8		dfdisj2 4367	. 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 367   A.wal 1403   e. wcel 1842   E*wmo 2239   C_ wss 3413  Disj wdisj 4365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-rmo 2761  df-in 3420  df-ss 3427  df-disj 4366

This theorem is referenced by:  disjeq1  4372  disjx0  4389  disjxiun  4391  disjss3  4393  volfiniun  22141  uniioovol  22172  uniioombllem4  22179  disjiunel  27768  carsggect  28646  carsgclctunlem2  28647  omsmeas  28651  sibfof  28668