unssin - Intuitionistic Logic Explorer

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Theorem unssin 3443

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 786	. . . . 5 $\/$ $/\$
2		eldifn 3327	. . . . . 6 $\$
3		eldifn 3327	. . . . . 6 $\$
4	2, 3	anim12i 338	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 631	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3387	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 681	. . 3 $\/$ $\$ $\$
8		elun 3345	. . 3 $\/$
9		vex 2802	. . . 4
10		eldif 3206	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 946	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 201	. 2 $\$ $\$ $\$
13	12	ssriv 3228	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104 $\/$ wo 713

wcel 2200

cvv 2799 $\$ cdif 3194

cun 3195

cin 3196

wss 3197

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-ext 2211

This theorem depends on definitions: df-bi 117 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-v 2801 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210

This theorem is referenced by: (None)

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