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Theorem 0ntr 21679
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 4280 . . . . 5 (𝑋𝑆 ↔ (𝑋𝑆) = ∅)
2 eqss 3933 . . . . . . . . 9 (𝑆 = 𝑋 ↔ (𝑆𝑋𝑋𝑆))
3 fveq2 6649 . . . . . . . . . . . . 13 (𝑆 = 𝑋 → ((int‘𝐽)‘𝑆) = ((int‘𝐽)‘𝑋))
4 clscld.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
54ntrtop 21678 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
63, 5sylan9eqr 2858 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → ((int‘𝐽)‘𝑆) = 𝑋)
76eqeq1d 2803 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ 𝑋 = ∅))
87biimpd 232 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))
98ex 416 . . . . . . . . 9 (𝐽 ∈ Top → (𝑆 = 𝑋 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
102, 9syl5bir 246 . . . . . . . 8 (𝐽 ∈ Top → ((𝑆𝑋𝑋𝑆) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
1110expd 419 . . . . . . 7 (𝐽 ∈ Top → (𝑆𝑋 → (𝑋𝑆 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))))
1211com34 91 . . . . . 6 (𝐽 ∈ Top → (𝑆𝑋 → (((int‘𝐽)‘𝑆) = ∅ → (𝑋𝑆𝑋 = ∅))))
1312imp32 422 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆𝑋 = ∅))
141, 13syl5bir 246 . . . 4 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → ((𝑋𝑆) = ∅ → 𝑋 = ∅))
1514necon3d 3011 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ≠ ∅ → (𝑋𝑆) ≠ ∅))
1615imp 410 . 2 (((𝐽 ∈ Top ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) ∧ 𝑋 ≠ ∅) → (𝑋𝑆) ≠ ∅)
1716an32s 651 1 (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  cdif 3881  wss 3884  c0 4246   cuni 4803  cfv 6328  Topctop 21501  intcnt 21625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-top 21502  df-ntr 21628
This theorem is referenced by: (None)
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