unssin - Intuitionistic Logic Explorer

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Theorem unssin 3470

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 793	. . . . 5 $\/$ $/\$
2		eldifn 3352	. . . . . 6 $\$
3		eldifn 3352	. . . . . 6 $\$
4	2, 3	anim12i 338	. . . . 5 $\$ $/\$ $\$ $/\$
5	1, 4	nsyl 637	. . . 4 $\/$ $\$ $/\$ $\$
6		elin 3412	. . . 4 $\$ $\$ $\$ $/\$ $\$
7	5, 6	sylnibr 688	. . 3 $\/$ $\$ $\$
8		elun 3370	. . 3 $\/$
9		vex 2824	. . . 4
10		eldif 3229	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
11	9, 10	mpbiran 953	. . 3 $\$ $\$ $\$ $\$ $\$
12	7, 8, 11	3imtr4i 201	. 2 $\$ $\$ $\$
13	12	ssriv 3252	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104 $\/$ wo 720

wcel 2209

cvv 2821 $\$ cdif 3217

cun 3218

cin 3219

wss 3220

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-ext 2220

This theorem depends on definitions: df-bi 117 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233

This theorem is referenced by: (None)

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