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Theorem opkabssvvk 4208

Description: Any Kuratowski ordered pair abstraction is a subset of

. (Contributed by SF, 13-Jan-2015.)

**Assertion**
Ref	Expression
opkabssvvk	$/\$ _k

(

)

Proof of Theorem opkabssvvk Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2353	. . . . . . 7
2		vex 2862	. . . . . . . 8
3		vex 2862	. . . . . . . 8
4		opkeq12 4061	. . . . . . . . 9 $/\$
5	4	eqeq2d 2364	. . . . . . . 8 $/\$
6	2, 3, 5	spc2ev 2947	. . . . . . 7
7	1, 6	ax-mp 8	. . . . . 6
8		elvvk 4207	. . . . . 6 _k
9	7, 8	mpbir 200	. . . . 5 _k
10		eleq1 2413	. . . . 5 _k _k
11	9, 10	mpbiri 224	. . . 4 _k
12	11	adantr 451	. . 3 $/\$ _k
13	12	exlimivv 1635	. 2 $/\$ _k
14	13	abssi 3341	1 $/\$ _k

Colors of variables: wff setvar class

Syntax hints: $/\$ wa 358

wex 1541

wceq 1642

wcel 1710

cab 2339

cvv 2859

wss 3257

copk 4057

_k cxpk 4174

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185

This theorem is referenced by: opkabssvvki 4209