Theorem ntrcls0 14799

Description: A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntrcls0	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅)

Proof of Theorem ntrcls0

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
2		clscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	clsss3 14798	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
4	2	sscls 14788	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
5	2	ntrss 14787	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋 ∧ 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
6	1, 3, 4, 5	syl3anc 1271	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
7	6	3adant3 1041	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
8		sseq2 3248	. . . 4 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅ → (((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((int‘𝐽)‘𝑆) ⊆ ∅))
9	8	3ad2ant3 1044	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → (((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((int‘𝐽)‘𝑆) ⊆ ∅))
10	7, 9	mpbid 147	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ∅)
11		ss0 3532	. 2 ⊢ (((int‘𝐽)‘𝑆) ⊆ ∅ → ((int‘𝐽)‘𝑆) = ∅)
12	10, 11	syl 14	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197  ∅c0 3491  ∪ cuni 3887  ‘cfv 5317  Topctop 14665  intcnt 14761  clsccl 14762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-top 14666  df-cld 14763  df-ntr 14764  df-cls 14765

This theorem is referenced by: (None)