Theorem opkelins3kg 4253

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

Assertion

Ref	Expression
opkelins3kg	|- ((A e. V /\ B e. W) -> (<<A, B>> e. Ins3k C <-> E.xE.yE.z(A = {{x}} /\ B = <<y, z>> /\ <<x, y>> e. C)))

Distinct variable groups:   x,A,y,z   x,B,y,z   x,C,y,z

Allowed substitution hints:   V(x,y,z)   W(x,y,z)

Proof of Theorem opkelins3kg

Dummy variables w t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ins3k 4189	. 2 |- Ins3k C = {t | E.wE.u(t = <<w, u>> /\ E.xE.yE.z(w = {{x}} /\ u = <<y, z>> /\ <<x, y>> e. C))}
2		eqeq1 2359	. . . 4 |- (w = A -> (w = {{x}} <-> A = {{x}}))
3	2	3anbi1d 1256	. . 3 |- (w = A -> ((w = {{x}} /\ u = <<y, z>> /\ <<x, y>> e. C) <-> (A = {{x}} /\ u = <<y, z>> /\ <<x, y>> e. C)))
4	3	3exbidv 1629	. 2 |- (w = A -> (E.xE.yE.z(w = {{x}} /\ u = <<y, z>> /\ <<x, y>> e. C) <-> E.xE.yE.z(A = {{x}} /\ u = <<y, z>> /\ <<x, y>> e. C)))
5		eqeq1 2359	. . . 4 |- (u = B -> (u = <<y, z>> <-> B = <<y, z>>))
6	5	3anbi2d 1257	. . 3 |- (u = B -> ((A = {{x}} /\ u = <<y, z>> /\ <<x, y>> e. C) <-> (A = {{x}} /\ B = <<y, z>> /\ <<x, y>> e. C)))
7	6	3exbidv 1629	. 2 |- (u = B -> (E.xE.yE.z(A = {{x}} /\ u = <<y, z>> /\ <<x, y>> e. C) <-> E.xE.yE.z(A = {{x}} /\ B = <<y, z>> /\ <<x, y>> e. C)))
8	1, 4, 7	opkelopkabg 4246	1 |- ((A e. V /\ B e. W) -> (<<A, B>> e. Ins3k C <-> E.xE.yE.z(A = {{x}} /\ B = <<y, z>> /\ <<x, y>> e. C)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  {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:  otkelins3kg  4255  ins3kss  4281