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Theorem refref 23238
Description: Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
Assertion
Ref Expression
refref (𝐴𝑉𝐴Ref𝐴)

Proof of Theorem refref
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . 3 𝐴 = 𝐴
2 ssid 4004 . . . . 5 𝑥𝑥
3 sseq2 4008 . . . . . 6 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
43rspcev 3612 . . . . 5 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
52, 4mpan2 688 . . . 4 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
65rgen 3062 . . 3 𝑥𝐴𝑦𝐴 𝑥𝑦
71, 6pm3.2i 470 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
81, 1isref 23234 . 2 (𝐴𝑉 → (𝐴Ref𝐴 ↔ ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
97, 8mpbiri 258 1 (𝐴𝑉𝐴Ref𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  wrex 3069  wss 3948   cuni 4908   class class class wbr 5148  Refcref 23227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-ref 23230
This theorem is referenced by:  locfinref  33120
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