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Theorem exse 5487
 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 5201 . . 3 (𝐴𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
21ralrimivw 3153 . 2 (𝐴𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
3 df-se 5483 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
42, 3sylibr 237 1 (𝐴𝑉𝑅 Se 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  ∀wral 3109  {crab 3113  Vcvv 3444   class class class wbr 5033   Se wse 5480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773  ax-sep 5170 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-se 5483 This theorem is referenced by:  wemoiso  7660  wemoiso2  7661  oiiso  8989  hartogslem1  8994  oemapwe  9145  cantnffval2  9146  om2uzoi  13322  uzsinds  13354  bpolylem  15397
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