![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ntr0 | Structured version Visualization version 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 22850 | . 2 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
2 | 0ss 4398 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
3 | eqid 2725 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 23014 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ 𝐽 ↔ ((int‘𝐽)‘∅) = ∅)) |
5 | 2, 4 | mpan2 689 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ 𝐽 ↔ ((int‘𝐽)‘∅) = ∅)) |
6 | 1, 5 | mpbid 231 | 1 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ∅c0 4322 ∪ cuni 4909 ‘cfv 6549 Topctop 22839 intcnt 22965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-top 22840 df-ntr 22968 |
This theorem is referenced by: iccntr 24781 |
Copyright terms: Public domain | W3C validator |