unssin - Intuitionistic Logic Explorer

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Theorem unssin 3398

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 782	. . . . 5 $\/$ $/\$
2		eldifn 3282	. . . . . 6 $\$
3		eldifn 3282	. . . . . 6 $\$
4	2, 3	anim12i 338	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 629	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3342	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 678	. . 3 $\/$ $\$ $\$
8		elun 3300	. . 3 $\/$
9		vex 2763	. . . 4
10		eldif 3162	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 942	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 201	. 2 $\$ $\$ $\$
13	12	ssriv 3183	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104 $\/$ wo 709

wcel 2164

cvv 2760 $\$ cdif 3150

cun 3151

cin 3152

wss 3153

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175

This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166

This theorem is referenced by: (None)

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