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Theorem exse 5644
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 5337 . . 3 (𝐴𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
21ralrimivw 3139 . 2 (𝐴𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
3 df-se 5637 . 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 2098  wral 3050  {crab 3418  Vcvv 3461   class class class wbr 5152   Se wse 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5303
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-in 3953  df-ss 3963  df-se 5637
This theorem is referenced by:  wemoiso  7986  wemoiso2  7987  oiiso  9576  hartogslem1  9581  oemapwe  9733  cantnffval2  9734  om2uzoi  13970  uzsinds  14002  bpolylem  16045  om2noseqoi  28269
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