Theorem sikssvvk 4266

Description: A Kuratowski singleton image is a Kuratowski relationship. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
sikssvvk	⊢ SIk A ⊆ (V ×k V)

Proof of Theorem sikssvvk

Dummy variables x y z t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sik 4192	. 2 ⊢ SIk A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ∃t∃u(y = {t} ∧ z = {u} ∧ ⟪t, u⟫ ∈ A))}
2	1	opkabssvvki 4209	1 ⊢ SIk A ⊆ (V ×k V)

Syntax hints:   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟪copk 4057   ×_k cxpk 4174   SI_k csik 4181

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-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  df-sik 4192

This theorem is referenced by:  sikss1c1c  4267