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Theorem sikss1c1c 4267
 Description: A Kuratowski singleton image is a subset of (1c ×k 1c). (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
sikss1c1c SIk A (1c ×k 1c)

Proof of Theorem sikss1c1c
Dummy variables x y z w t a b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sik 4192 . . . . 5 SIk A = {t zw(t = ⟪z, w ab(z = {a} w = {b} a, b A))}
2 eqeq1 2359 . . . . . . 7 (z = x → (z = {a} ↔ x = {a}))
323anbi1d 1256 . . . . . 6 (z = x → ((z = {a} w = {b} a, b A) ↔ (x = {a} w = {b} a, b A)))
432exbidv 1628 . . . . 5 (z = x → (ab(z = {a} w = {b} a, b A) ↔ ab(x = {a} w = {b} a, b A)))
5 eqeq1 2359 . . . . . . 7 (w = y → (w = {b} ↔ y = {b}))
653anbi2d 1257 . . . . . 6 (w = y → ((x = {a} w = {b} a, b A) ↔ (x = {a} y = {b} a, b A)))
762exbidv 1628 . . . . 5 (w = y → (ab(x = {a} w = {b} a, b A) ↔ ab(x = {a} y = {b} a, b A)))
8 vex 2862 . . . . 5 x V
9 vex 2862 . . . . 5 y V
101, 4, 7, 8, 9opkelopkab 4246 . . . 4 (⟪x, y SIk Aab(x = {a} y = {b} a, b A))
11 opkeq12 4061 . . . . . . 7 ((x = {a} y = {b}) → ⟪x, y⟫ = ⟪{a}, {b}⟫)
12 vex 2862 . . . . . . . . 9 a V
1312snel1c 4140 . . . . . . . 8 {a} 1c
14 vex 2862 . . . . . . . . 9 b V
1514snel1c 4140 . . . . . . . 8 {b} 1c
16 opkelxpkg 4247 . . . . . . . . 9 (({a} 1c {b} 1c) → (⟪{a}, {b}⟫ (1c ×k 1c) ↔ ({a} 1c {b} 1c)))
1713, 15, 16mp2an 653 . . . . . . . 8 (⟪{a}, {b}⟫ (1c ×k 1c) ↔ ({a} 1c {b} 1c))
1813, 15, 17mpbir2an 886 . . . . . . 7 ⟪{a}, {b}⟫ (1c ×k 1c)
1911, 18syl6eqel 2441 . . . . . 6 ((x = {a} y = {b}) → ⟪x, y (1c ×k 1c))
20193adant3 975 . . . . 5 ((x = {a} y = {b} a, b A) → ⟪x, y (1c ×k 1c))
2120exlimivv 1635 . . . 4 (ab(x = {a} y = {b} a, b A) → ⟪x, y (1c ×k 1c))
2210, 21sylbi 187 . . 3 (⟪x, y SIk A → ⟪x, y (1c ×k 1c))
2322gen2 1547 . 2 xy(⟪x, y SIk A → ⟪x, y (1c ×k 1c))
24 sikssvvk 4266 . . 3 SIk A (V ×k V)
25 ssrelk 4211 . . 3 ( SIk A (V ×k V) → ( SIk A (1c ×k 1c) ↔ xy(⟪x, y SIk A → ⟪x, y (1c ×k 1c))))
2624, 25ax-mp 8 . 2 ( SIk A (1c ×k 1c) ↔ xy(⟪x, y SIk A → ⟪x, y (1c ×k 1c)))
2723, 26mpbir 200 1 SIk A (1c ×k 1c)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟪copk 4057  1cc1c 4134   ×k cxpk 4174   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-ral 2619  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-1c 4136  df-xpk 4185  df-sik 4192 This theorem is referenced by:  opkelimagekg  4271  sikexg  4296  dfnnc2  4395
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