unssin - Intuitionistic Logic Explorer

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Theorem unssin 3420

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 3304	. . . . . 6 $\$
3		eldifn 3304	. . . . . 6 $\$
4	2, 3	anim12i 338	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 629	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3364	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 679	. . 3 $\/$ $\$ $\$
8		elun 3322	. . 3 $\/$
9		vex 2779	. . . 4
10		eldif 3183	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 943	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 201	. 2 $\$ $\$ $\$
13	12	ssriv 3205	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

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

wcel 2178

cvv 2776 $\$ cdif 3171

cun 3172

cin 3173

wss 3174

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 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-ext 2189

This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-v 2778 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187

This theorem is referenced by: (None)

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