unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3368

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 2738	. . . . . . . 8
2		eldif 3136	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 940	. . . . . . 7 $\$
4	3	anbi1i 458	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3136	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 752	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 212	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 120	. . . 4 $\$ $\$ $\/$
9	8	con2i 627	. . 3 $\/$ $\$ $\$
10		elun 3274	. . 3 $\/$
11		eldif 3136	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 940	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 201	. 2 $\$ $\$ $\$
14	13	ssriv 3157	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104 $\/$ wo 708

wcel 2146

cvv 2735 $\$ cdif 3124

cun 3125

wss 3127

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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-ext 2157

This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-v 2737 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140

This theorem is referenced by: (None)