unssin - Intuitionistic Logic Explorer

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Theorem unssin 3176

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 807	. . . . 5 $\/$ $/\$
2		eldifn 3067	. . . . . 6 $\$
3		eldifn 3067	. . . . . 6 $\$
4	2, 3	anim12i 321	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 558	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3126	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 602	. . 3 $\/$ $\$ $\$
8		elun 3084	. . 3 $\/$
9		vex 2560	. . . 4
10		eldif 2927	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 847	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 190	. 2 $\$ $\$ $\$
13	12	ssriv 2949	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 97 $\/$ wo 629

wcel 1393

cvv 2557 $\$ cdif 2914

cun 2915

cin 2916

wss 2917

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022

This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931

This theorem is referenced by: (None)

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