Theorem disjss2 3909

Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjss2	|- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )

Proof of Theorem disjss2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3091	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
2	1	ralimi 2495	. . . 4 |- ( A. x e. A B C_ C -> A. x e. A ( y e. B -> y e. C ) )
3		rmoim 2885	. . . 4 |- ( A. x e. A ( y e. B -> y e. C ) -> ( E* x e. A y e. C -> E* x e. A y e. B ) )
4	2, 3	syl 14	. . 3 |- ( A. x e. A B C_ C -> ( E* x e. A y e. C -> E* x e. A y e. B ) )
5	4	alimdv 1851	. 2 |- ( A. x e. A B C_ C -> ( A. y E* x e. A y e. C -> A. y E* x e. A y e. B ) )
6		df-disj 3907	. 2 |- (Disj x e. A C <-> A. y E* x e. A y e. C )
7		df-disj 3907	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
8	5, 6, 7	3imtr4g 204	1 |- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )

Syntax hints:   -> wi 4   A.wal 1329   e. wcel 1480   A.wral 2416   E*wrmo 2419   C_ wss 3071  Disj wdisj 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-rmo 2424  df-in 3077  df-ss 3084  df-disj 3907

This theorem is referenced by:  disjeq2  3910  0disj  3926