Theorem disjss2 3748

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 2939	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
2	1	ralimi 2384	. . . 4 |- ( A. x e. A B C_ C -> A. x e. A ( y e. B -> y e. C ) )
3		rmoim 2740	. . . 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 1759	. 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 3746	. 2 |- (Disj x e. A C <-> A. y E* x e. A y e. C )
7		df-disj 3746	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
8	5, 6, 7	3imtr4g 194	1 |- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )

Syntax hints:   -> wi 4   A.wal 1241   e. wcel 1393   A.wral 2306   E*wrmo 2309   C_ wss 2917  Disj wdisj 3745

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rmo 2314  df-in 2924  df-ss 2931  df-disj 3746

This theorem is referenced by:  disjeq2  3749  0disj  3761