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Theorem isfne3 33804
Description: The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne3 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦
Allowed substitution hints:   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem isfne3
StepHypRef Expression
1 isfne.1 . . 3 𝑋 = 𝐴
2 isfne.2 . . 3 𝑌 = 𝐵
31, 2isfne4 33801 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
4 dfss3 3903 . . . 4 (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵))
5 eltg3 21567 . . . . 5 (𝐵𝐶 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
65ralbidv 3162 . . . 4 (𝐵𝐶 → (∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦)))
74, 6syl5bb 286 . . 3 (𝐵𝐶 → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦)))
87anbi2d 631 . 2 (𝐵𝐶 → ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
93, 8syl5bb 286 1 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2111  wral 3106  wss 3881   cuni 4800   class class class wbr 5030  cfv 6324  topGenctg 16703  Fnecfne 33797
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-topgen 16709  df-fne 33798
This theorem is referenced by: (None)
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