unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3399

Description: Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)

**Assertion**
Ref	Expression
unssdif	$\$ $\$ $\$

Proof of Theorem unssdif Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2766	. . . . . . . 8
2		eldif 3166	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 942	. . . . . . 7 $\$
4	3	anbi1i 458	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3166	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 753	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 212	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 120	. . . 4 $\$ $\$ $\/$
9	8	con2i 628	. . 3 $\/$ $\$ $\$
10		elun 3305	. . 3 $\/$
11		eldif 3166	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 942	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 201	. 2 $\$ $\$ $\$
14	13	ssriv 3188	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

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

wcel 2167

cvv 2763 $\$ cdif 3154

cun 3155

wss 3157

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-ext 2178

This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170

This theorem is referenced by: (None)