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Theorem exse 5636
Description: Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
exse (𝐴𝑉𝑅 Se 𝐴)

Proof of Theorem exse
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabexg 5328 . . 3 (𝐴𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
21ralrimivw 3146 . 2 (𝐴𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
3 df-se 5629 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
42, 3sylibr 233 1 (𝐴𝑉𝑅 Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wral 3057  {crab 3428  Vcvv 3470   class class class wbr 5143   Se wse 5626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rab 3429  df-v 3472  df-in 3952  df-ss 3962  df-se 5629
This theorem is referenced by:  wemoiso  7972  wemoiso2  7973  oiiso  9555  hartogslem1  9560  oemapwe  9712  cantnffval2  9713  om2uzoi  13947  uzsinds  13979  bpolylem  16019  om2noseqoi  28170
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