Theorem inssun 3362

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		pm3.1 744	. . . . 5 |- ( ( x e. A /\ x e. B ) -> -. ( -. x e. A \/ -. x e. B ) )
2		eldifn 3245	. . . . . 6 |- ( x e. ( _V \ A ) -> -. x e. A )
3		eldifn 3245	. . . . . 6 |- ( x e. ( _V \ B ) -> -. x e. B )
4	2, 3	orim12i 749	. . . . 5 |- ( ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) -> ( -. x e. A \/ -. x e. B ) )
5	1, 4	nsyl 618	. . . 4 |- ( ( x e. A /\ x e. B ) -> -. ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
6		elun 3263	. . . 4 |- ( x e. ( ( _V \ A ) u. ( _V \ B ) ) <-> ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
7	5, 6	sylnibr 667	. . 3 |- ( ( x e. A /\ x e. B ) -> -. x e. ( ( _V \ A ) u. ( _V \ B ) ) )
8		elin 3305	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
9		vex 2729	. . . 4 |- x e. _V
10		eldif 3125	. . . 4 |- ( x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) <-> ( x e. _V /\ -. x e. ( ( _V \ A ) u. ( _V \ B ) ) ) )
11	9, 10	mpbiran 930	. . 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 3146	1 |- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 698   e. wcel 2136   _Vcvv 2726   \ cdif 3113   u. cun 3114   i^i cin 3115   C_ wss 3116

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 604  ax-in2 605  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-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by: (None)