unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3311

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 2689	. . . . . . . 8
2		eldif 3080	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 924	. . . . . . 7 $\$
4	3	anbi1i 453	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3080	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 741	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 211	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 119	. . . 4 $\$ $\$ $\/$
9	8	con2i 616	. . 3 $\/$ $\$ $\$
10		elun 3217	. . 3 $\/$
11		eldif 3080	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 924	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 200	. 2 $\$ $\$ $\$
14	13	ssriv 3101	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 697

wcel 1480

cvv 2686 $\$ cdif 3068

cun 3069

wss 3071

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121

This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084

This theorem is referenced by: (None)