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Theorem 0ntr 23094
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 4371 . . . . 5 (𝑋𝑆 ↔ (𝑋𝑆) = ∅)
2 eqss 4010 . . . . . . . . 9 (𝑆 = 𝑋 ↔ (𝑆𝑋𝑋𝑆))
3 fveq2 6906 . . . . . . . . . . . . 13 (𝑆 = 𝑋 → ((int‘𝐽)‘𝑆) = ((int‘𝐽)‘𝑋))
4 clscld.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
54ntrtop 23093 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
63, 5sylan9eqr 2796 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → ((int‘𝐽)‘𝑆) = 𝑋)
76eqeq1d 2736 . . . . . . . . . . 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 2958 . . 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 1536  wcel 2105  wne 2937  cdif 3959  wss 3962  c0 4338   cuni 4911  cfv 6562  Topctop 22914  intcnt 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-ntr 23043
This theorem is referenced by: (None)
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