difxp - Metamath Proof Explorer

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Theorem difxp 5558

Description: Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)

**Assertion**
Ref	Expression
difxp	$\$ $\$ $\$

Proof of Theorem difxp Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 3737	. . 3 $\$
2		relxp 5227	. . 3
3		relss 5206	. . 3 $\$ $\$
4	1, 2, 3	mp2 9	. 2 $\$
5		relxp 5227	. . 3 $\$
6		relxp 5227	. . 3 $\$
7		relun 5235	. . 3 $\$ $\$ $\$ $/\$ $\$
8	5, 6, 7	mpbir2an 955	. 2 $\$ $\$
9		ianor 509	. . . . . 6 $/\$ $\/$
10	9	anbi2i 730	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
11		andi 911	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$
12	10, 11	bitri 264	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
13		opelxp 5146	. . . . 5 $/\$
14		opelxp 5146	. . . . . 6 $/\$
15	14	notbii 310	. . . . 5 $/\$
16	13, 15	anbi12i 733	. . . 4 $/\$ $/\$ $/\$ $/\$
17		opelxp 5146	. . . . . 6 $\$ $\$ $/\$
18		eldif 3584	. . . . . . . 8 $\$ $/\$
19	18	anbi1i 731	. . . . . . 7 $\$ $/\$ $/\$ $/\$
20		an32 839	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	19, 20	bitri 264	. . . . . 6 $\$ $/\$ $/\$ $/\$
22	17, 21	bitri 264	. . . . 5 $\$ $/\$ $/\$
23		eldif 3584	. . . . . . 7 $\$ $/\$
24	23	anbi2i 730	. . . . . 6 $/\$ $\$ $/\$ $/\$
25		opelxp 5146	. . . . . 6 $\$ $/\$ $\$
26		anass 681	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27	24, 25, 26	3bitr4i 292	. . . . 5 $\$ $/\$ $/\$
28	22, 27	orbi12i 543	. . . 4 $\$ $\/$ $\$ $/\$ $/\$ $\/$ $/\$ $/\$
29	12, 16, 28	3bitr4i 292	. . 3 $/\$ $\$ $\/$ $\$
30		eldif 3584	. . 3 $\$ $/\$
31		elun 3753	. . 3 $\$ $\$ $\$ $\/$ $\$
32	29, 30, 31	3bitr4i 292	. 2 $\$ $\$ $\$
33	4, 8, 32	eqrelriiv 5214	1 $\$ $\$ $\$

Colors of variables: wff setvar class

Syntax hints:

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

wceq 1483

wcel 1990 $\$ cdif 3571

cun 3572

wss 3574

cop 4183

cxp 5112

wrel 5119

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 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 df-rel 5121

This theorem is referenced by: difxp1 5559 difxp2 5560 evlslem4 19508 txcld 21406

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