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Theorem fneref 36351
Description: Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
Assertion
Ref Expression
fneref (𝐴𝑉𝐴Fne𝐴)

Proof of Theorem fneref
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 𝐴 = 𝐴
2 ssid 4006 . . . . 5 𝑥𝑥
3 elequ2 2123 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
4 sseq1 4009 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
53, 4anbi12d 632 . . . . . 6 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
65rspcev 3622 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
72, 6mpanr2 704 . . . 4 ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
87rgen2 3199 . . 3 𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥)
91, 8pm3.2i 470 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))
101, 1isfne2 36343 . 2 (𝐴𝑉 → (𝐴Fne𝐴 ↔ ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))))
119, 10mpbiri 258 1 (𝐴𝑉𝐴Fne𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   cuni 4907   class class class wbr 5143  Fnecfne 36337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-topgen 17488  df-fne 36338
This theorem is referenced by: (None)
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