dfun2 - Metamath Proof Explorer

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Theorem dfun2 3859

Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3860 and dfss4 3858 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation $\$ (class difference). (Contributed by NM, 10-Jun-2004.)

**Assertion**
Ref	Expression
dfun2	$\$ $\$ $\$

Proof of Theorem dfun2 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . 7
2		eldif 3584	. . . . . . 7 $\$ $/\$
3	1, 2	mpbiran 953	. . . . . 6 $\$
4	3	anbi1i 731	. . . . 5 $\$ $/\$ $/\$
5		eldif 3584	. . . . 5 $\$ $\$ $\$ $/\$
6		ioran 511	. . . . 5 $\/$ $/\$
7	4, 5, 6	3bitr4i 292	. . . 4 $\$ $\$ $\/$
8	7	con2bii 347	. . 3 $\/$ $\$ $\$
9		eldif 3584	. . . 4 $\$ $\$ $\$ $/\$ $\$ $\$
10	1, 9	mpbiran 953	. . 3 $\$ $\$ $\$ $\$ $\$
11	8, 10	bitr4i 267	. 2 $\/$ $\$ $\$ $\$
12	11	uneqri 3755	1 $\$ $\$ $\$

Colors of variables: wff setvar class

Syntax hints:

wn 3 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

cvv 3200 $\$ cdif 3571

cun 3572

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579

This theorem is referenced by: dfun3 3865 dfin3 3866

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