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Mirrors > Home > NFE Home > Th. List > opkelidkg | GIF version |
Description: Membership in the Kuratowski identity relationship. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
opkelidkg | ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ Ik ↔ A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-idk 4195 | . 2 ⊢ Ik = {z ∣ ∃x∃y(z = ⟪x, y⟫ ∧ x = y)} | |
2 | eqeq1 2359 | . 2 ⊢ (x = A → (x = y ↔ A = y)) | |
3 | eqeq2 2362 | . 2 ⊢ (y = B → (A = y ↔ A = B)) | |
4 | 1, 2, 3 | opkelopkabg 4245 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ Ik ↔ A = B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ⟪copk 4057 Ik cidk 4184 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-idk 4195 |
This theorem is referenced by: dfidk2 4313 nnsucelrlem1 4424 nndisjeq 4429 eqtfinrelk 4486 oddfinex 4504 evenodddisjlem1 4515 dfphi2 4569 |
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