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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

Proof of Theorem dfidk2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idkssvvk 4281 . 2 k k
2 inss1 3475 . . 3 Sk k Sk Sk
3 ssetkssvvk 4278 . . 3 Sk k
42, 3sstri 3281 . 2 Sk k Sk k
5 eqss 3287 . . 3
6 vex 2862 . . . 4
7 vex 2862 . . . 4
8 opkelidkg 4274 . . . 4 k
96, 7, 8mp2an 653 . . 3 k
10 elin 3219 . . . 4 Sk k Sk Sk k Sk
11 opkelssetkg 4268 . . . . . 6 Sk
126, 7, 11mp2an 653 . . . . 5 Sk
136, 7opkelcnvk 4250 . . . . . 6 k Sk Sk
14 opkelssetkg 4268 . . . . . . 7 Sk
157, 6, 14mp2an 653 . . . . . 6 Sk
1613, 15bitri 240 . . . . 5 k Sk
1712, 16anbi12i 678 . . . 4 Sk k Sk
1810, 17bitri 240 . . 3 Sk k Sk
195, 9, 183bitr4i 268 . 2 k Sk k Sk
201, 4, 19eqrelkriiv 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-mp 5  ax-1 6  ax-2 7  ax-3 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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