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Theorem fneref 34635
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 2736 . . 3 𝐴 = 𝐴
2 ssid 3954 . . . . 5 𝑥𝑥
3 elequ2 2120 . . . . . . 7 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
4 sseq1 3957 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
53, 4anbi12d 631 . . . . . 6 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
65rspcev 3570 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
72, 6mpanr2 701 . . . 4 ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐴 (𝑦𝑧𝑧𝑥))
87rgen2 3190 . . 3 𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥)
91, 8pm3.2i 471 . 2 ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))
101, 1isfne2 34627 . 2 (𝐴𝑉 → (𝐴Fne𝐴 ↔ ( 𝐴 = 𝐴 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐴 (𝑦𝑧𝑧𝑥))))
119, 10mpbiri 257 1 (𝐴𝑉𝐴Fne𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  wral 3061  wrex 3070  wss 3898   cuni 4852   class class class wbr 5092  Fnecfne 34621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-topgen 17251  df-fne 34622
This theorem is referenced by: (None)
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