Theorem opkelssetkg 4269

Description: Membership in the Kuratowski subset relationship. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
opkelssetkg	|- ((A e. V /\ B e. W) -> (<<A, B>> e. Sk <-> A C_ B))

Proof of Theorem opkelssetkg

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

Step	Hyp	Ref	Expression
1		df-ssetk 4194	. 2 |- Sk = {x | E.yE.z(x = <<y, z>> /\ y C_ z)}
2		sseq1 3293	. 2 |- (y = A -> (y C_ z <-> A C_ z))
3		sseq2 3294	. 2 |- (z = B -> (A C_ z <-> A C_ B))
4	1, 2, 3	opkelopkabg 4246	1 |- ((A e. V /\ B e. W) -> (<<A, B>> e. Sk <-> A C_ B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710   C_ wss 3258  <<copk 4058   S_k cssetk 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 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-ssetk 4194

This theorem is referenced by:  elssetkg  4270  ssetkex  4295  dfidk2  4314  ssfin  4471  eqpwrelk  4479  srelk  4525  dfsset2  4744