Theorem disjss2 3969

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 3141	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
2	1	ralimi 2533	. . . 4 |- ( A. x e. A B C_ C -> A. x e. A ( y e. B -> y e. C ) )
3		rmoim 2931	. . . 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 1872	. 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 3967	. 2 |- (Disj x e. A C <-> A. y E* x e. A y e. C )
7		df-disj 3967	. 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 1346   e. wcel 2141   A.wral 2448   E*wrmo 2451   C_ wss 3121  Disj wdisj 3966

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-rmo 2456  df-in 3127  df-ss 3134  df-disj 3967

This theorem is referenced by:  disjeq2  3970  0disj  3986