Theorem inssun 3471

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		pm3.1 766	. . . . 5 |- ( ( x e. A /\ x e. B ) -> -. ( -. x e. A \/ -. x e. B ) )
2		eldifn 3352	. . . . . 6 |- ( x e. ( _V \ A ) -> -. x e. A )
3		eldifn 3352	. . . . . 6 |- ( x e. ( _V \ B ) -> -. x e. B )
4	2, 3	orim12i 771	. . . . 5 |- ( ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) -> ( -. x e. A \/ -. x e. B ) )
5	1, 4	nsyl 637	. . . 4 |- ( ( x e. A /\ x e. B ) -> -. ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
6		elun 3370	. . . 4 |- ( x e. ( ( _V \ A ) u. ( _V \ B ) ) <-> ( x e. ( _V \ A ) \/ x e. ( _V \ B ) ) )
7	5, 6	sylnibr 688	. . 3 |- ( ( x e. A /\ x e. B ) -> -. x e. ( ( _V \ A ) u. ( _V \ B ) ) )
8		elin 3412	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
9		vex 2824	. . . 4 |- x e. _V
10		eldif 3229	. . . 4 |- ( x e. ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) <-> ( x e. _V /\ -. x e. ( ( _V \ A ) u. ( _V \ B ) ) ) )
11	9, 10	mpbiran 953	. . 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 3252	1 |- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 720   e. wcel 2209   _Vcvv 2821   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by: (None)