MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ntr Structured version   Visualization version   GIF version

Theorem 0ntr 23079
Description: A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
0ntr (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆) ≠ ∅)

Proof of Theorem 0ntr
StepHypRef Expression
1 ssdif0 4366 . . . . 5 (𝑋𝑆 ↔ (𝑋𝑆) = ∅)
2 eqss 3999 . . . . . . . . 9 (𝑆 = 𝑋 ↔ (𝑆𝑋𝑋𝑆))
3 fveq2 6906 . . . . . . . . . . . . 13 (𝑆 = 𝑋 → ((int‘𝐽)‘𝑆) = ((int‘𝐽)‘𝑋))
4 clscld.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
54ntrtop 23078 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
63, 5sylan9eqr 2799 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → ((int‘𝐽)‘𝑆) = 𝑋)
76eqeq1d 2739 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ 𝑋 = ∅))
87biimpd 229 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))
98ex 412 . . . . . . . . 9 (𝐽 ∈ Top → (𝑆 = 𝑋 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
102, 9biimtrrid 243 . . . . . . . 8 (𝐽 ∈ Top → ((𝑆𝑋𝑋𝑆) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
1110expd 415 . . . . . . 7 (𝐽 ∈ Top → (𝑆𝑋 → (𝑋𝑆 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))))
1211com34 91 . . . . . 6 (𝐽 ∈ Top → (𝑆𝑋 → (((int‘𝐽)‘𝑆) = ∅ → (𝑋𝑆𝑋 = ∅))))
1312imp32 418 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆𝑋 = ∅))
141, 13biimtrrid 243 . . . 4 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → ((𝑋𝑆) = ∅ → 𝑋 = ∅))
1514necon3d 2961 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ≠ ∅ → (𝑋𝑆) ≠ ∅))
1615imp 406 . 2 (((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) ∧ 𝑋 ≠ ∅) → (𝑋𝑆) ≠ ∅)
1716an32s 652 1 (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  cdif 3948  wss 3951  c0 4333   cuni 4907  cfv 6561  Topctop 22899  intcnt 23025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-ntr 23028
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator