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Theorem cldopn 12315
Description: The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cldopn (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)

Proof of Theorem cldopn
StepHypRef Expression
1 cldrcl 12310 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 12311 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simplbda 382 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋𝑆) ∈ 𝐽)
51, 4mpancom 419 1 (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481  cdif 3073  wss 3076   cuni 3744  cfv 5131  Topctop 12203  Clsdccld 12300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-top 12204  df-cld 12303
This theorem is referenced by:  difopn  12316  uncld  12321  cnclima  12431
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