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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 13171 | . 2 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
2 | 0ss 3461 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
3 | eqid 2177 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 13292 | . . 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 1353 ∈ wcel 2148 ⊆ wss 3129 ∅c0 3422 ∪ cuni 3807 ‘cfv 5212 Topctop 13162 intcnt 13260 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-top 13163 df-ntr 13263 |
This theorem is referenced by: (None) |
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