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Theorem refref 22887
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 2733 . . 3 𝐴 = 𝐴
2 ssid 3970 . . . . 5 𝑥𝑥
3 sseq2 3974 . . . . . 6 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
43rspcev 3583 . . . . 5 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
52, 4mpan2 690 . . . 4 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
65rgen 3063 . . 3 𝑥𝐴𝑦𝐴 𝑥𝑦
71, 6pm3.2i 472 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
81, 1isref 22883 . 2 (𝐴𝑉 → (𝐴Ref𝐴 ↔ ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
97, 8mpbiri 258 1 (𝐴𝑉𝐴Ref𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  wss 3914   cuni 4869   class class class wbr 5109  Refcref 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-ref 22879
This theorem is referenced by:  locfinref  32486
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