unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3439

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 2802	. . . . . . . 8
2		eldif 3206	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 946	. . . . . . 7 $\$
4	3	anbi1i 458	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3206	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 757	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 212	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 120	. . . 4 $\$ $\$ $\/$
9	8	con2i 630	. . 3 $\/$ $\$ $\$
10		elun 3345	. . 3 $\/$
11		eldif 3206	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 946	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 201	. 2 $\$ $\$ $\$
14	13	ssriv 3228	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104 $\/$ wo 713

wcel 2200

cvv 2799 $\$ cdif 3194

cun 3195

wss 3197

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-ext 2211

This theorem depends on definitions: df-bi 117 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-v 2801 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210

This theorem is referenced by: (None)