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Theorem fnetr 33687
Description: Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
fnetr ((𝐴Fne𝐵𝐵Fne𝐶) → 𝐴Fne𝐶)

Proof of Theorem fnetr
StepHypRef Expression
1 eqid 2819 . . . 4 𝐴 = 𝐴
2 eqid 2819 . . . 4 𝐵 = 𝐵
31, 2fnebas 33680 . . 3 (𝐴Fne𝐵 𝐴 = 𝐵)
4 eqid 2819 . . . 4 𝐶 = 𝐶
52, 4fnebas 33680 . . 3 (𝐵Fne𝐶 𝐵 = 𝐶)
63, 5sylan9eq 2874 . 2 ((𝐴Fne𝐵𝐵Fne𝐶) → 𝐴 = 𝐶)
7 fnerel 33674 . . . . 5 Rel Fne
87brrelex2i 5602 . . . 4 (𝐴Fne𝐵𝐵 ∈ V)
91, 2isfne4b 33677 . . . . 5 (𝐵 ∈ V → (𝐴Fne𝐵 ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
109simplbda 502 . . . 4 ((𝐵 ∈ V ∧ 𝐴Fne𝐵) → (topGen‘𝐴) ⊆ (topGen‘𝐵))
118, 10mpancom 686 . . 3 (𝐴Fne𝐵 → (topGen‘𝐴) ⊆ (topGen‘𝐵))
127brrelex2i 5602 . . . 4 (𝐵Fne𝐶𝐶 ∈ V)
132, 4isfne4b 33677 . . . . 5 (𝐶 ∈ V → (𝐵Fne𝐶 ↔ ( 𝐵 = 𝐶 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐶))))
1413simplbda 502 . . . 4 ((𝐶 ∈ V ∧ 𝐵Fne𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
1512, 14mpancom 686 . . 3 (𝐵Fne𝐶 → (topGen‘𝐵) ⊆ (topGen‘𝐶))
1611, 15sylan9ss 3978 . 2 ((𝐴Fne𝐵𝐵Fne𝐶) → (topGen‘𝐴) ⊆ (topGen‘𝐶))
1712adantl 484 . . 3 ((𝐴Fne𝐵𝐵Fne𝐶) → 𝐶 ∈ V)
181, 4isfne4b 33677 . . 3 (𝐶 ∈ V → (𝐴Fne𝐶 ↔ ( 𝐴 = 𝐶 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐶))))
1917, 18syl 17 . 2 ((𝐴Fne𝐵𝐵Fne𝐶) → (𝐴Fne𝐶 ↔ ( 𝐴 = 𝐶 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐶))))
206, 16, 19mpbir2and 711 1 ((𝐴Fne𝐵𝐵Fne𝐶) → 𝐴Fne𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  wss 3934   cuni 4830   class class class wbr 5057  cfv 6348  topGenctg 16703  Fnecfne 33672
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-topgen 16709  df-fne 33673
This theorem is referenced by:  fnessref  33693  fnemeet2  33703  fnejoin2  33705
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