unssin - Intuitionistic Logic Explorer

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Theorem unssin 3366

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 776	. . . . 5 $\/$ $/\$
2		eldifn 3250	. . . . . 6 $\$
3		eldifn 3250	. . . . . 6 $\$
4	2, 3	anim12i 336	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 623	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3310	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 672	. . 3 $\/$ $\$ $\$
8		elun 3268	. . 3 $\/$
9		vex 2733	. . . 4
10		eldif 3130	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 935	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 200	. 2 $\$ $\$ $\$
13	12	ssriv 3151	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 703

wcel 2141

cvv 2730 $\$ cdif 3118

cun 3119

cin 3120

wss 3121

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152

This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134

This theorem is referenced by: (None)

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