unssdif - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > unssdif		Unicode version

Theorem unssdif 3362

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 2733	. . . . . . . 8
2		eldif 3130	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 935	. . . . . . 7 $\$
4	3	anbi1i 455	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3130	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 747	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 211	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 119	. . . 4 $\$ $\$ $\/$
9	8	con2i 622	. . 3 $\/$ $\$ $\$
10		elun 3268	. . 3 $\/$
11		eldif 3130	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 935	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 200	. 2 $\$ $\$ $\$
14	13	ssriv 3151	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 703

wcel 2141

cvv 2730 $\$ cdif 3118

cun 3119

wss 3121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152

This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134

This theorem is referenced by: (None)