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Theorem fneref 35743
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 2726 . . 3 𝐴 = 𝐴
2 ssid 3999 . . . . 5 𝑥𝑥
3 elequ2 2113 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
4 sseq1 4002 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
53, 4anbi12d 630 . . . . . 6 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
65rspcev 3606 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
72, 6mpanr2 701 . . . 4 ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
87rgen2 3191 . . 3 𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥)
91, 8pm3.2i 470 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))
101, 1isfne2 35735 . 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 1533  wcel 2098  wral 3055  wrex 3064  wss 3943   cuni 4902   class class class wbr 5141  Fnecfne 35729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-topgen 17396  df-fne 35730
This theorem is referenced by: (None)
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