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Theorem refref 21126
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 2610 . . 3 𝐴 = 𝐴
2 ssid 3587 . . . . 5 𝑥𝑥
3 sseq2 3590 . . . . . 6 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
43rspcev 3282 . . . . 5 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
52, 4mpan2 703 . . . 4 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
65rgen 2906 . . 3 𝑥𝐴𝑦𝐴 𝑥𝑦
71, 6pm3.2i 470 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
81, 1isref 21122 . 2 (𝐴𝑉 → (𝐴Ref𝐴 ↔ ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
97, 8mpbiri 247 1 (𝐴𝑉𝐴Ref𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  wss 3540   cuni 4372   class class class wbr 4583  Refcref 21115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-ref 21118
This theorem is referenced by:  locfinref  29236
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