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Mirrors > Home > NFE Home > Th. List > opksnelsik | GIF version |
Description: Membership of an ordered pair of singletons in a Kuratowski singleton image. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
opksnelsik.1 | ⊢ A ∈ V |
opksnelsik.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
opksnelsik | ⊢ (⟪{A}, {B}⟫ ∈ SIk C ↔ ⟪A, B⟫ ∈ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 4111 | . . 3 ⊢ {A} ∈ V | |
2 | snex 4111 | . . 3 ⊢ {B} ∈ V | |
3 | opkelsikg 4264 | . . 3 ⊢ (({A} ∈ V ∧ {B} ∈ V) → (⟪{A}, {B}⟫ ∈ SIk C ↔ ∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C))) | |
4 | 1, 2, 3 | mp2an 653 | . 2 ⊢ (⟪{A}, {B}⟫ ∈ SIk C ↔ ∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C)) |
5 | eqcom 2355 | . . . . . 6 ⊢ ({A} = {x} ↔ {x} = {A}) | |
6 | vex 2862 | . . . . . . 7 ⊢ x ∈ V | |
7 | 6 | sneqb 3876 | . . . . . 6 ⊢ ({x} = {A} ↔ x = A) |
8 | 5, 7 | bitri 240 | . . . . 5 ⊢ ({A} = {x} ↔ x = A) |
9 | eqcom 2355 | . . . . . 6 ⊢ ({B} = {y} ↔ {y} = {B}) | |
10 | vex 2862 | . . . . . . 7 ⊢ y ∈ V | |
11 | 10 | sneqb 3876 | . . . . . 6 ⊢ ({y} = {B} ↔ y = B) |
12 | 9, 11 | bitri 240 | . . . . 5 ⊢ ({B} = {y} ↔ y = B) |
13 | biid 227 | . . . . 5 ⊢ (⟪x, y⟫ ∈ C ↔ ⟪x, y⟫ ∈ C) | |
14 | 8, 12, 13 | 3anbi123i 1140 | . . . 4 ⊢ (({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C) ↔ (x = A ∧ y = B ∧ ⟪x, y⟫ ∈ C)) |
15 | 14 | 2exbii 1583 | . . 3 ⊢ (∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C) ↔ ∃x∃y(x = A ∧ y = B ∧ ⟪x, y⟫ ∈ C)) |
16 | opksnelsik.1 | . . . 4 ⊢ A ∈ V | |
17 | opksnelsik.2 | . . . 4 ⊢ B ∈ V | |
18 | opkeq1 4059 | . . . . 5 ⊢ (x = A → ⟪x, y⟫ = ⟪A, y⟫) | |
19 | 18 | eleq1d 2419 | . . . 4 ⊢ (x = A → (⟪x, y⟫ ∈ C ↔ ⟪A, y⟫ ∈ C)) |
20 | opkeq2 4060 | . . . . 5 ⊢ (y = B → ⟪A, y⟫ = ⟪A, B⟫) | |
21 | 20 | eleq1d 2419 | . . . 4 ⊢ (y = B → (⟪A, y⟫ ∈ C ↔ ⟪A, B⟫ ∈ C)) |
22 | 16, 17, 19, 21 | ceqsex2v 2896 | . . 3 ⊢ (∃x∃y(x = A ∧ y = B ∧ ⟪x, y⟫ ∈ C) ↔ ⟪A, B⟫ ∈ C) |
23 | 15, 22 | bitri 240 | . 2 ⊢ (∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C) ↔ ⟪A, B⟫ ∈ C) |
24 | 4, 23 | bitri 240 | 1 ⊢ (⟪{A}, {B}⟫ ∈ SIk C ↔ ⟪A, B⟫ ∈ C) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ w3a 934 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 {csn 3737 ⟪copk 4057 SIk csik 4181 |
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 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: opkelimagekg 4271 sikexg 4296 dfimak2 4298 dfaddc2 4381 dfnnc2 4395 nnsucelrlem1 4424 ltfinex 4464 eqpwrelk 4478 eqpw1relk 4479 ncfinraiselem2 4480 ncfinlowerlem1 4482 eqtfinrelk 4486 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnadjoinlem1 4519 nnpweqlem1 4522 srelk 4524 sfintfinlem1 4531 tfinnnlem1 4533 spfinex 4537 setconslem2 4732 setconslem3 4733 setconslem7 4737 dfswap2 4741 |
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