unssin - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > unssin		Unicode version

Theorem unssin 3412

Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)

**Assertion**
Ref	Expression
unssin	$\$ $\$ $\$

Proof of Theorem unssin Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oranim 783	. . . . 5 $\/$ $/\$
2		eldifn 3296	. . . . . 6 $\$
3		eldifn 3296	. . . . . 6 $\$
4	2, 3	anim12i 338	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 629	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3356	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 679	. . 3 $\/$ $\$ $\$
8		elun 3314	. . 3 $\/$
9		vex 2775	. . . 4
10		eldif 3175	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 943	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 201	. 2 $\$ $\$ $\$
13	12	ssriv 3197	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104 $\/$ wo 710

wcel 2176

cvv 2772 $\$ cdif 3163

cun 3164

cin 3165

wss 3166

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187

This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179

This theorem is referenced by: (None)

W3C validator