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Theorem refref 22049
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 2818 . . 3 𝐴 = 𝐴
2 ssid 3986 . . . . 5 𝑥𝑥
3 sseq2 3990 . . . . . 6 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
43rspcev 3620 . . . . 5 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
52, 4mpan2 687 . . . 4 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
65rgen 3145 . . 3 𝑥𝐴𝑦𝐴 𝑥𝑦
71, 6pm3.2i 471 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
81, 1isref 22045 . 2 (𝐴𝑉 → (𝐴Ref𝐴 ↔ ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
97, 8mpbiri 259 1 (𝐴𝑉𝐴Ref𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933   cuni 4830   class class class wbr 5057  Refcref 22038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-ref 22041
This theorem is referenced by:  locfinref  31004
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