unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3172

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 2560	. . . . . . . 8
2		eldif 2927	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 847	. . . . . . 7 $\$
4	3	anbi1i 431	. . . . . 6 $\$ $/\$ $/\$
5		eldif 2927	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 669	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 201	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 113	. . . 4 $\$ $\$ $\/$
9	8	con2i 557	. . 3 $\/$ $\$ $\$
10		elun 3084	. . 3 $\/$
11		eldif 2927	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 847	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 190	. 2 $\$ $\$ $\$
14	13	ssriv 2949	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 97 $\/$ wo 629

wcel 1393

cvv 2557 $\$ cdif 2914

cun 2915

wss 2917

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022

This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931

This theorem is referenced by: (None)