inssun - Intuitionistic Logic Explorer

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Theorem inssun 3347

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	$\$ $\$ $\$

Proof of Theorem inssun Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.1 744	. . . . 5 $/\$ $\/$
2		eldifn 3230	. . . . . 6 $\$
3		eldifn 3230	. . . . . 6 $\$
4	2, 3	orim12i 749	. . . . 5 $\$ $\/$ $\$ $\/$
5	1, 4	nsyl 618	. . . 4 $/\$ $\$ $\/$ $\$
6		elun 3248	. . . 4 $\$ $\$ $\$ $\/$ $\$
7	5, 6	sylnibr 667	. . 3 $/\$ $\$ $\$
8		elin 3290	. . 3 $/\$
9		vex 2715	. . . 4
10		eldif 3111	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 925	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 200	. 2 $\$ $\$ $\$
13	12	ssriv 3132	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 698

wcel 2128

cvv 2712 $\$ cdif 3099

cun 3100

cin 3101

wss 3102

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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139

This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115

This theorem is referenced by: (None)

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