Theorem disjss2 3985

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 3151	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
2	1	ralimi 2540	. . . 4 |- ( A. x e. A B C_ C -> A. x e. A ( y e. B -> y e. C ) )
3		rmoim 2940	. . . 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 1879	. 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 3983	. 2 |- (Disj x e. A C <-> A. y E* x e. A y e. C )
7		df-disj 3983	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
8	5, 6, 7	3imtr4g 205	1 |- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )

Syntax hints:   -> wi 4   A.wal 1351   e. wcel 2148   A.wral 2455   E*wrmo 2458   C_ wss 3131  Disj wdisj 3982

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rmo 2463  df-in 3137  df-ss 3144  df-disj 3983

This theorem is referenced by:  disjeq2  3986  0disj  4002