Theorem inssun 3311

Description: Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)

Assertion

Ref	Expression
inssun	|- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )

Proof of Theorem inssun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.1 743	. . . . 5 |- ( ( x e. A /\ x e. B ) -> -. ( -. x e. A \/ -. x e. B ) )
2		eldifn 3194	. . . . . 6 |- ( x e. ( _V \ A ) -> -. x e. A )
3		eldifn 3194	. . . . . 6 |- ( x e. ( _V \ B ) -> -. x e. B )
4	2, 3	orim12i 748	. . . . 5 |- ( ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) -> ( -. x e. A \/ -. x e. B ) )
5	1, 4	nsyl 617	. . . 4 |- ( ( x e. A /\ x e. B ) -> -. ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
6		elun 3212	. . . 4 |- ( x e. ( ( _V \ A ) u. ( _V \ B ) ) <-> ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
7	5, 6	sylnibr 666	. . 3 |- ( ( x e. A /\ x e. B ) -> -. x e. ( ( _V \ A ) u. ( _V \ B ) ) )
8		elin 3254	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
9		vex 2684	. . . 4 |- x e. _V
10		eldif 3075	. . . 4 |- ( x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) <-> ( x e. _V /\ -. x e. ( ( _V \ A ) u. ( _V \ B ) ) ) )
11	9, 10	mpbiran 924	. . 3 |- ( x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) <-> -. x e. ( ( _V \ A ) u. ( _V \ B ) ) )
12	7, 8, 11	3imtr4i 200	. 2 |- ( x e. ( A i^i B ) -> x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) )
13	12	ssriv 3096	1 |- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 697   e. wcel 1480   _Vcvv 2681   \ cdif 3063   u. cun 3064   i^i cin 3065   C_ wss 3066

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-in1 603  ax-in2 604  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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079

This theorem is referenced by: (None)