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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 13983 | . 2 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
2 | 0ss 3476 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
3 | eqid 2189 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 14102 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ 𝐽 ↔ ((int‘𝐽)‘∅) = ∅)) |
5 | 2, 4 | mpan2 425 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ 𝐽 ↔ ((int‘𝐽)‘∅) = ∅)) |
6 | 1, 5 | mpbid 147 | 1 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 ∅c0 3437 ∪ cuni 3824 ‘cfv 5235 Topctop 13974 intcnt 14070 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-top 13975 df-ntr 14073 |
This theorem is referenced by: (None) |
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