Theorem inssun 3412

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		pm3.1 755	. . . . 5 |- ( ( x e. A /\ x e. B ) -> -. ( -. x e. A \/ -. x e. B ) )
2		eldifn 3295	. . . . . 6 |- ( x e. ( _V \ A ) -> -. x e. A )
3		eldifn 3295	. . . . . 6 |- ( x e. ( _V \ B ) -> -. x e. B )
4	2, 3	orim12i 760	. . . . 5 |- ( ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) -> ( -. x e. A \/ -. x e. B ) )
5	1, 4	nsyl 629	. . . 4 |- ( ( x e. A /\ x e. B ) -> -. ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
6		elun 3313	. . . 4 |- ( x e. ( ( _V \ A ) u. ( _V \ B ) ) <-> ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
7	5, 6	sylnibr 678	. . 3 |- ( ( x e. A /\ x e. B ) -> -. x e. ( ( _V \ A ) u. ( _V \ B ) ) )
8		elin 3355	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
9		vex 2774	. . . 4 |- x e. _V
10		eldif 3174	. . . 4 |- ( x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) <-> ( x e. _V /\ -. x e. ( ( _V \ A ) u. ( _V \ B ) ) ) )
11	9, 10	mpbiran 942	. . 3 |- ( x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) <-> -. x e. ( ( _V \ A ) u. ( _V \ B ) ) )
12	7, 8, 11	3imtr4i 201	. 2 |- ( x e. ( A i^i B ) -> x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) )
13	12	ssriv 3196	1 |- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 709   e. wcel 2175   _Vcvv 2771   \ cdif 3162   u. cun 3163   i^i cin 3164   C_ wss 3165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178

This theorem is referenced by: (None)