unssin - Intuitionistic Logic Explorer

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Theorem unssin 3361

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 771	. . . . 5 $\/$ $/\$
2		eldifn 3245	. . . . . 6 $\$
3		eldifn 3245	. . . . . 6 $\$
4	2, 3	anim12i 336	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 618	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3305	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 667	. . 3 $\/$ $\$ $\$
8		elun 3263	. . 3 $\/$
9		vex 2729	. . . 4
10		eldif 3125	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 930	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 200	. 2 $\$ $\$ $\$
13	12	ssriv 3146	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 698

wcel 2136

cvv 2726 $\$ cdif 3113

cun 3114

cin 3115

wss 3116

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147

This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129

This theorem is referenced by: (None)

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