unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3357

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 2729	. . . . . . . 8
2		eldif 3125	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 930	. . . . . . 7 $\$
4	3	anbi1i 454	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3125	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 742	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 211	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 119	. . . 4 $\$ $\$ $\/$
9	8	con2i 617	. . 3 $\/$ $\$ $\$
10		elun 3263	. . 3 $\/$
11		eldif 3125	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 930	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 200	. 2 $\$ $\$ $\$
14	13	ssriv 3146	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103 $\/$ wo 698

wcel 2136

cvv 2726 $\$ cdif 3113

cun 3114

wss 3116

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147

This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129

This theorem is referenced by: (None)