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Mirrors > Home > ILE Home > Th. List > ntr0 | GIF version |
Description: The interior of the empty set. (Contributed by NM, 2-Oct-2007.) |
Ref | Expression |
---|---|
ntr0 | ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0opn 12010 | . 2 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
2 | 0ss 3365 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
3 | eqid 2113 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 12131 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ 𝐽 ↔ ((int‘𝐽)‘∅) = ∅)) |
5 | 2, 4 | mpan2 419 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ 𝐽 ↔ ((int‘𝐽)‘∅) = ∅)) |
6 | 1, 5 | mpbid 146 | 1 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ∈ wcel 1461 ⊆ wss 3035 ∅c0 3327 ∪ cuni 3700 ‘cfv 5079 Topctop 12001 intcnt 12099 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-top 12002 df-ntr 12102 |
This theorem is referenced by: (None) |
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