unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3258

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 2644	. . . . . . . 8
2		eldif 3030	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 892	. . . . . . 7 $\$
4	3	anbi1i 449	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3030	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 710	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 211	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 119	. . . 4 $\$ $\$ $\/$
9	8	con2i 597	. . 3 $\/$ $\$ $\$
10		elun 3164	. . 3 $\/$
11		eldif 3030	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 892	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 200	. 2 $\$ $\$ $\$
14	13	ssriv 3051	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 670

wcel 1448

cvv 2641 $\$ cdif 3018

cun 3019

wss 3021

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082

This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034

This theorem is referenced by: (None)