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Theorem dfidk2 4313

Description: Definition of k in terms of Sk. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
dfidk2	k Sk k Sk

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Proof of Theorem dfidk2 Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idkssvvk 4281	. 2 k k
2		inss1 3475	. . 3 Sk k Sk Sk
3		ssetkssvvk 4278	. . 3 Sk k
4	2, 3	sstri 3281	. 2 Sk k Sk k
5		eqss 3287	. . 3
6		vex 2862	. . . 4
7		vex 2862	. . . 4
8		opkelidkg 4274	. . . 4 k
9	6, 7, 8	mp2an 653	. . 3 k
10		elin 3219	. . . 4 Sk k Sk Sk k Sk
11		opkelssetkg 4268	. . . . . 6 Sk
12	6, 7, 11	mp2an 653	. . . . 5 Sk
13	6, 7	opkelcnvk 4250	. . . . . 6 k Sk Sk
14		opkelssetkg 4268	. . . . . . 7 Sk
15	7, 6, 14	mp2an 653	. . . . . 6 Sk
16	13, 15	bitri 240	. . . . 5 k Sk
17	12, 16	anbi12i 678	. . . 4 Sk k Sk
18	10, 17	bitri 240	. . 3 Sk k Sk
19	5, 9, 18	3bitr4i 268	. 2 k Sk k Sk
20	1, 4, 19	eqrelkriiv 4213	1 k Sk k Sk

Colors of variables: wff setvar class

Syntax hints:   wb 176   wa 358   wceq 1642   wcel 1710  cvv 2859   cin 3208   wss 3257  copk 4057   k cxpk 4174  kccnvk 4175   Sk cssetk 4183   k cidk 4184

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-3an 936  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-ral 2619  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  df-cnvk 4186  df-ssetk 4193  df-idk 4195

This theorem is referenced by:  idkex  4314

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