unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3199

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 2604	. . . . . . . 8
2		eldif 2982	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 881	. . . . . . 7 $\$
4	3	anbi1i 445	. . . . . 6 $\$ $/\$ $/\$
5		eldif 2982	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 701	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 210	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 118	. . . 4 $\$ $\$ $\/$
9	8	con2i 589	. . 3 $\/$ $\$ $\$
10		elun 3113	. . 3 $\/$
11		eldif 2982	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 881	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 199	. 2 $\$ $\$ $\$
14	13	ssriv 3003	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 102 $\/$ wo 661

wcel 1433

cvv 2601 $\$ cdif 2970

cun 2971

wss 2973

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063

This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986

This theorem is referenced by: (None)