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Mirrors > Home > NFE Home > Th. List > opkelsikg | GIF version |
Description: Membership in Kuratowski singleton image. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
opkelsikg | ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ SIk C ↔ ∃x∃y(A = {x} ∧ B = {y} ∧ ⟪x, y⟫ ∈ C))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sik 4192 | . 2 ⊢ SIk C = {z ∣ ∃t∃u(z = ⟪t, u⟫ ∧ ∃x∃y(t = {x} ∧ u = {y} ∧ ⟪x, y⟫ ∈ C))} | |
2 | eqeq1 2359 | . . . 4 ⊢ (t = A → (t = {x} ↔ A = {x})) | |
3 | 2 | 3anbi1d 1256 | . . 3 ⊢ (t = A → ((t = {x} ∧ u = {y} ∧ ⟪x, y⟫ ∈ C) ↔ (A = {x} ∧ u = {y} ∧ ⟪x, y⟫ ∈ C))) |
4 | 3 | 2exbidv 1628 | . 2 ⊢ (t = A → (∃x∃y(t = {x} ∧ u = {y} ∧ ⟪x, y⟫ ∈ C) ↔ ∃x∃y(A = {x} ∧ u = {y} ∧ ⟪x, y⟫ ∈ C))) |
5 | eqeq1 2359 | . . . 4 ⊢ (u = B → (u = {y} ↔ B = {y})) | |
6 | 5 | 3anbi2d 1257 | . . 3 ⊢ (u = B → ((A = {x} ∧ u = {y} ∧ ⟪x, y⟫ ∈ C) ↔ (A = {x} ∧ B = {y} ∧ ⟪x, y⟫ ∈ C))) |
7 | 6 | 2exbidv 1628 | . 2 ⊢ (u = B → (∃x∃y(A = {x} ∧ u = {y} ∧ ⟪x, y⟫ ∈ C) ↔ ∃x∃y(A = {x} ∧ B = {y} ∧ ⟪x, y⟫ ∈ C))) |
8 | 1, 4, 7 | opkelopkabg 4245 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ SIk C ↔ ∃x∃y(A = {x} ∧ B = {y} ∧ ⟪x, y⟫ ∈ C))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {csn 3737 ⟪copk 4057 SIk csik 4181 |
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-sik 4192 |
This theorem is referenced by: opksnelsik 4265 dfpw12 4301 setconslem1 4731 dfsi2 4751 |
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