Theorem disjss2 3962

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 3136	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
2	1	ralimi 2529	. . . 4 |- ( A. x e. A B C_ C -> A. x e. A ( y e. B -> y e. C ) )
3		rmoim 2927	. . . 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 1867	. 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 3960	. 2 |- (Disj x e. A C <-> A. y E* x e. A y e. C )
7		df-disj 3960	. 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 1341   e. wcel 2136   A.wral 2444   E*wrmo 2447   C_ wss 3116  Disj wdisj 3959

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-rmo 2452  df-in 3122  df-ss 3129  df-disj 3960

This theorem is referenced by:  disjeq2  3963  0disj  3979