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Theorem 0ntr 21089
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 4089 . . . . 5 (𝑋𝑆 ↔ (𝑋𝑆) = ∅)
2 eqss 3767 . . . . . . . . 9 (𝑆 = 𝑋 ↔ (𝑆𝑋𝑋𝑆))
3 fveq2 6330 . . . . . . . . . . . . 13 (𝑆 = 𝑋 → ((int‘𝐽)‘𝑆) = ((int‘𝐽)‘𝑋))
4 clscld.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
54ntrtop 21088 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
63, 5sylan9eqr 2827 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → ((int‘𝐽)‘𝑆) = 𝑋)
76eqeq1d 2773 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ 𝑋 = ∅))
87biimpd 219 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))
98ex 397 . . . . . . . . 9 (𝐽 ∈ Top → (𝑆 = 𝑋 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
102, 9syl5bir 233 . . . . . . . 8 (𝐽 ∈ Top → ((𝑆𝑋𝑋𝑆) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
1110expd 400 . . . . . . 7 (𝐽 ∈ Top → (𝑆𝑋 → (𝑋𝑆 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))))
1211com34 91 . . . . . 6 (𝐽 ∈ Top → (𝑆𝑋 → (((int‘𝐽)‘𝑆) = ∅ → (𝑋𝑆𝑋 = ∅))))
1312imp32 405 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆𝑋 = ∅))
141, 13syl5bir 233 . . . 4 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → ((𝑋𝑆) = ∅ → 𝑋 = ∅))
1514necon3d 2964 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ≠ ∅ → (𝑋𝑆) ≠ ∅))
1615imp 393 . 2 (((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) ∧ 𝑋 ≠ ∅) → (𝑋𝑆) ≠ ∅)
1716an32s 631 1 (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  cdif 3720  wss 3723  c0 4063   cuni 4574  cfv 6029  Topctop 20911  intcnt 21035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-top 20912  df-ntr 21038
This theorem is referenced by: (None)
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