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Theorem setswith 4321
 Description: Two ways to express the class of all sets that contain A. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
setswith {x A x} = if(A V, ( Skk {{A}}), )
Distinct variable group:   x,A

Proof of Theorem setswith
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 snex 4111 . . . . . . 7 {A} V
2 opkeq1 4059 . . . . . . . 8 (y = {A} → ⟪y, x⟫ = ⟪{A}, x⟫)
32eleq1d 2419 . . . . . . 7 (y = {A} → (⟪y, x Sk ↔ ⟪{A}, x Sk ))
41, 3rexsn 3768 . . . . . 6 (y {{A}}⟪y, x Sk ↔ ⟪{A}, x Sk )
5 vex 2862 . . . . . . 7 x V
6 elssetkg 4269 . . . . . . 7 ((A V x V) → (⟪{A}, x SkA x))
75, 6mpan2 652 . . . . . 6 (A V → (⟪{A}, x SkA x))
84, 7syl5rbb 249 . . . . 5 (A V → (A xy {{A}}⟪y, x Sk ))
98abbidv 2467 . . . 4 (A V → {x A x} = {x y {{A}}⟪y, x Sk })
10 df-imak 4189 . . . 4 ( Skk {{A}}) = {x y {{A}}⟪y, x Sk }
119, 10syl6eqr 2403 . . 3 (A V → {x A x} = ( Skk {{A}}))
12 iftrue 3668 . . 3 (A V → if(A V, ( Skk {{A}}), ) = ( Skk {{A}}))
1311, 12eqtr4d 2388 . 2 (A V → {x A x} = if(A V, ( Skk {{A}}), ))
14 elex 2867 . . . . . 6 (A xA V)
1514con3i 127 . . . . 5 A V → ¬ A x)
1615alrimiv 1631 . . . 4 A V → x ¬ A x)
17 ab0 3569 . . . 4 ({x A x} = x ¬ A x)
1816, 17sylibr 203 . . 3 A V → {x A x} = )
19 iffalse 3669 . . 3 A V → if(A V, ( Skk {{A}}), ) = )
2018, 19eqtr4d 2388 . 2 A V → {x A x} = if(A V, ( Skk {{A}}), ))
2113, 20pm2.61i 156 1 {x A x} = if(A V, ( Skk {{A}}), )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859  ∅c0 3550   ifcif 3662  {csn 3737  ⟪copk 4057   “k cimak 4179   Sk cssetk 4183 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-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663  df-sn 3741  df-pr 3742  df-opk 4058  df-imak 4189  df-ssetk 4193 This theorem is referenced by:  setswithex  4322
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