Theorem opkelins3kg 4253

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

Assertion

Ref	Expression
opkelins3kg	⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ Ins3k C ↔ ∃x∃y∃z(A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ 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 ∣ ∃w∃u(t = ⟪w, u⟫ ∧ ∃x∃y∃z(w = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ 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⟫ ∈ C) ↔ (A = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ C)))
4	3	3exbidv 1629	. 2 ⊢ (w = A → (∃x∃y∃z(w = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ C) ↔ ∃x∃y∃z(A = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ 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⟫ ∈ C) ↔ (A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ C)))
7	6	3exbidv 1629	. 2 ⊢ (u = B → (∃x∃y∃z(A = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ C) ↔ ∃x∃y∃z(A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ C)))
8	1, 4, 7	opkelopkabg 4246	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ Ins3k C ↔ ∃x∃y∃z(A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ C)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ 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