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Theorem fneref 35834
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 2728 . . 3 𝐴 = 𝐴
2 ssid 4002 . . . . 5 𝑥𝑥
3 elequ2 2114 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
4 sseq1 4005 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
53, 4anbi12d 631 . . . . . 6 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
65rspcev 3609 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
72, 6mpanr2 703 . . . 4 ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
87rgen2 3194 . . 3 𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥)
91, 8pm3.2i 470 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))
101, 1isfne2 35826 . 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 1534  wcel 2099  wral 3058  wrex 3067  wss 3947   cuni 4908   class class class wbr 5148  Fnecfne 35820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-topgen 17425  df-fne 35821
This theorem is referenced by: (None)
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