Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isfne2 Structured version   Visualization version   GIF version

Theorem isfne2 32664
Description: The predicate "𝐵 is finer than 𝐴." (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne2 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧
Allowed substitution hints:   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem isfne2
StepHypRef Expression
1 isfne.1 . . 3 𝑋 = 𝐴
2 isfne.2 . . 3 𝑌 = 𝐵
31, 2isfne4 32662 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
4 dfss3 3733 . . . 4 (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵))
5 eltg2b 20985 . . . . 5 (𝐵𝐶 → (𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
65ralbidv 3124 . . . 4 (𝐵𝐶 → (∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
74, 6syl5bb 272 . . 3 (𝐵𝐶 → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
87anbi2d 742 . 2 (𝐵𝐶 → ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
93, 8syl5bb 272 1 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715   cuni 4588   class class class wbr 4804  cfv 6049  topGenctg 16320  Fnecfne 32658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-topgen 16326  df-fne 32659
This theorem is referenced by:  fness  32671  fneref  32672  fnessref  32679
  Copyright terms: Public domain W3C validator