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Theorem refref 23535
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 2734 . . 3 𝐴 = 𝐴
2 ssid 4025 . . . . 5 𝑥𝑥
3 sseq2 4029 . . . . . 6 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
43rspcev 3631 . . . . 5 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
52, 4mpan2 690 . . . 4 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
65rgen 3065 . . 3 𝑥𝐴𝑦𝐴 𝑥𝑦
71, 6pm3.2i 470 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
81, 1isref 23531 . 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 1537  wcel 2103  wral 3063  wrex 3072  wss 3970   cuni 4931   class class class wbr 5169  Refcref 23524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-ref 23527
This theorem is referenced by:  locfinref  33779
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