unssdif - Intuitionistic Logic Explorer

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Theorem unssdif 3466

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 2824	. . . . . . . 8
2		eldif 3229	. . . . . . . 8 $\$ $/\$
3	1, 2	mpbiran 953	. . . . . . 7 $\$
4	3	anbi1i 462	. . . . . 6 $\$ $/\$ $/\$
5		eldif 3229	. . . . . 6 $\$ $\$ $\$ $/\$
6		ioran 764	. . . . . 6 $\/$ $/\$
7	4, 5, 6	3bitr4i 212	. . . . 5 $\$ $\$ $\/$
8	7	biimpi 120	. . . 4 $\$ $\$ $\/$
9	8	con2i 636	. . 3 $\/$ $\$ $\$
10		elun 3370	. . 3 $\/$
11		eldif 3229	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
12	1, 11	mpbiran 953	. . 3 $\$ $\$ $\$ $\$ $\$
13	9, 10, 12	3imtr4i 201	. 2 $\$ $\$ $\$
14	13	ssriv 3252	1 $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104 $\/$ wo 720

wcel 2209

cvv 2821 $\$ cdif 3217

cun 3218

wss 3220

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-ext 2220

This theorem depends on definitions: df-bi 117 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233

This theorem is referenced by: (None)