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 ≠ ∅) ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ∖ 𝑆) ≠ ∅)

Proof of Theorem 0ntr

Step	Hyp	Ref	Expression
1		ssdif0 4323	. . . . 5 ⊢ (𝑋 ⊆ 𝑆 ↔ (𝑋 ∖ 𝑆) = ∅)
2		eqss 3982	. . . . . . . . 9 ⊢ (𝑆 = 𝑋 ↔ (𝑆 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑆))
3		fveq2 6670	. . . . . . . . . . . . 13 ⊢ (𝑆 = 𝑋 → ((int‘𝐽)‘𝑆) = ((int‘𝐽)‘𝑋))
4		clscld.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
5	4	ntrtop 21678	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
6	3, 5	sylan9eqr 2878	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → ((int‘𝐽)‘𝑆) = 𝑋)
7	6	eqeq1d 2823	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ 𝑋 = ∅))
8	7	biimpd 231	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))
9	8	ex 415	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → (𝑆 = 𝑋 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
10	2, 9	syl5bir 245	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((𝑆 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑆) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))
11	10	expd 418	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 → (𝑋 ⊆ 𝑆 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))))
12	11	com34 91	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 → (((int‘𝐽)‘𝑆) = ∅ → (𝑋 ⊆ 𝑆 → 𝑋 = ∅))))
13	12	imp32 421	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ⊆ 𝑆 → 𝑋 = ∅))
14	1, 13	syl5bir 245	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → ((𝑋 ∖ 𝑆) = ∅ → 𝑋 = ∅))
15	14	necon3d 3037	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ≠ ∅ → (𝑋 ∖ 𝑆) ≠ ∅))
16	15	imp 409	. 2 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) ∧ 𝑋 ≠ ∅) → (𝑋 ∖ 𝑆) ≠ ∅)
17	16	an32s 650	1 ⊢ (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ∖ 𝑆) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4838  ‘cfv 6355  Topctop 21501  intcnt 21625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-top 21502  df-ntr 21628

This theorem is referenced by: (None)