inssun - Intuitionistic Logic Explorer

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Theorem inssun 3321

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 3204	. . . . . 6 $\$
3		eldifn 3204	. . . . . 6 $\$
4	2, 3	orim12i 749	. . . . 5 $\$ $\/$ $\$ $\/$
5	1, 4	nsyl 618	. . . 4 $/\$ $\$ $\/$ $\$
6		elun 3222	. . . 4 $\$ $\$ $\$ $\/$ $\$
7	5, 6	sylnibr 667	. . 3 $/\$ $\$ $\$
8		elin 3264	. . 3 $/\$
9		vex 2692	. . . 4
10		eldif 3085	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 925	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 200	. 2 $\$ $\$ $\$
13	12	ssriv 3106	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

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

wcel 1481

cvv 2689 $\$ cdif 3073

cun 3074

cin 3075

wss 3076

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122

This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089

This theorem is referenced by: (None)

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