Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isfne4b Structured version   Visualization version   GIF version

Theorem isfne4b 32642
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 479 . . . . . . 7 ((𝐵𝑉𝑋 = 𝑌) → 𝑋 = 𝑌)
2 isfne.1 . . . . . . 7 𝑋 = 𝐴
3 isfne.2 . . . . . . 7 𝑌 = 𝐵
41, 2, 33eqtr3g 2817 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 = 𝐵)
5 uniexg 7120 . . . . . . 7 (𝐵𝑉 𝐵 ∈ V)
65adantr 472 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐵 ∈ V)
74, 6eqeltrd 2839 . . . . 5 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
8 uniexb 7138 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8sylibr 224 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
10 simpl 474 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐵𝑉)
11 tgss3 20992 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
129, 10, 11syl2anc 696 . . 3 ((𝐵𝑉𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
1312pm5.32da 676 . 2 (𝐵𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵))))
142, 3isfne4 32641 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
1513, 14syl6rbbr 279 1 (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715   cuni 4588   class class class wbr 4804  cfv 6049  topGenctg 16300  Fnecfne 32637
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 7114
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-ne 2933  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-iun 4674  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 16306  df-fne 32638
This theorem is referenced by:  fnetr  32652  fneval  32653
  Copyright terms: Public domain W3C validator