Theorem otkelins3k 4257

Description: Kuratowski ordered triple membership in Kuratowski insertion operator. (Contributed by SF, 12-Jan-2015.)

Hypotheses

Ref	Expression
otkelinsk.1	|- A e. _V
otkelinsk.2	|- B e. _V
otkelinsk.3	|- C e. _V

Assertion

Ref	Expression
otkelins3k	|- (<<{{A}}, <<B, C>>>> e. Ins3k D <-> <<A, B>> e. D)

Proof of Theorem otkelins3k

Step	Hyp	Ref	Expression
1		otkelinsk.1	. 2 |- A e. _V
2		otkelinsk.2	. 2 |- B e. _V
3		otkelinsk.3	. 2 |- C e. _V
4		otkelins3kg 4255	. 2 |- ((A e. _V /\ B e. _V /\ C e. _V) -> (<<{{A}}, <<B, C>>>> e. Ins3k D <-> <<A, B>> e. D))
5	1, 2, 3, 4	mp3an 1277	1 |- (<<{{A}}, <<B, C>>>> e. Ins3k D <-> <<A, B>> e. D)

Syntax hints:   <-> wb 176   e. wcel 1710  _Vcvv 2860  {csn 3738  <<copk 4058   Ins3_k cins3k 4178

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-ins3k 4189

This theorem is referenced by:  ins3kexg  4307  ndisjrelk  4324  dfaddc2  4382  nnsucelrlem1  4425  ltfinex  4465  eqpwrelk  4479  eqpw1relk  4480  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  srelk  4525  sfintfinlem1  4532  tfinnnlem1  4534  spfinex  4538  dfop2lem1  4574  setconslem2  4733  setconslem3  4734  setconslem7  4738  dfswap2  4742