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Theorem fneref 33772
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 2822 . . 3 𝐴 = 𝐴
2 ssid 3964 . . . . 5 𝑥𝑥
3 elequ2 2129 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
4 sseq1 3967 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
53, 4anbi12d 633 . . . . . 6 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
65rspcev 3598 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
72, 6mpanr2 703 . . . 4 ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
87rgen2 3193 . . 3 𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥)
91, 8pm3.2i 474 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))
101, 1isfne2 33764 . 2 (𝐴𝑉 → (𝐴Fne𝐴 ↔ ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))))
119, 10mpbiri 261 1 (𝐴𝑉𝐴Fne𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  wrex 3131  wss 3908   cuni 4813   class class class wbr 5042  Fnecfne 33758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-topgen 16708  df-fne 33759
This theorem is referenced by: (None)
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