Theorem otkelins3k 4256

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

Hypotheses

Ref	Expression
otkelinsk.1	⊢ A ∈ V
otkelinsk.2	⊢ B ∈ V
otkelinsk.3	⊢ C ∈ V

Assertion

Ref	Expression
otkelins3k	⊢ (⟪{{A}}, ⟪B, C⟫⟫ ∈ Ins3k D ↔ ⟪A, B⟫ ∈ D)

Proof of Theorem otkelins3k

Step	Hyp	Ref	Expression
1		otkelinsk.1	. 2 ⊢ A ∈ V
2		otkelinsk.2	. 2 ⊢ B ∈ V
3		otkelinsk.3	. 2 ⊢ C ∈ V
4		otkelins3kg 4254	. 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (⟪{{A}}, ⟪B, C⟫⟫ ∈ Ins3k D ↔ ⟪A, B⟫ ∈ D))
5	1, 2, 3, 4	mp3an 1277	1 ⊢ (⟪{{A}}, ⟪B, C⟫⟫ ∈ Ins3k D ↔ ⟪A, B⟫ ∈ D)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2859  {csn 3737  ⟪copk 4057   Ins3_k cins3k 4177

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-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-ins3k 4188

This theorem is referenced by:  ins3kexg  4306  ndisjrelk  4323  dfaddc2  4381  nnsucelrlem1  4424  ltfinex  4464  eqpwrelk  4478  eqpw1relk  4479  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  nnpweqlem1  4522  srelk  4524  sfintfinlem1  4531  tfinnnlem1  4533  spfinex  4537  dfop2lem1  4573  setconslem2  4732  setconslem3  4733  setconslem7  4737  dfswap2  4741