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Theorem isfne4b 31506
 Description: A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne4b (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))

Proof of Theorem isfne4b
StepHypRef Expression
1 simpr 476 . . . . . . 7 ((𝐵𝑉𝑋 = 𝑌) → 𝑋 = 𝑌)
2 isfne.1 . . . . . . 7 𝑋 = 𝐴
3 isfne.2 . . . . . . 7 𝑌 = 𝐵
41, 2, 33eqtr3g 2667 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 = 𝐵)
5 uniexg 6853 . . . . . . 7 (𝐵𝑉 𝐵 ∈ V)
65adantr 480 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐵 ∈ V)
74, 6eqeltrd 2688 . . . . 5 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
8 uniexb 6866 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8sylibr 223 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
10 simpl 472 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐵𝑉)
11 tgss3 20601 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
129, 10, 11syl2anc 691 . . 3 ((𝐵𝑉𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
1312pm5.32da 671 . 2 (𝐵𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵))))
142, 3isfne4 31505 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
1513, 14syl6rbbr 278 1 (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  ‘cfv 5804  topGenctg 15921  Fnecfne 31501 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927  df-fne 31502 This theorem is referenced by:  fnetr  31516  fneval  31517
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