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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 22041 | . 2 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
2 | 0ss 4331 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
3 | eqid 2738 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 22205 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ 𝐽 ↔ ((int‘𝐽)‘∅) = ∅)) |
5 | 2, 4 | mpan2 688 | . 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 1539 ∈ wcel 2106 ⊆ wss 3887 ∅c0 4257 ∪ cuni 4840 ‘cfv 6427 Topctop 22030 intcnt 22156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-top 22031 df-ntr 22159 |
This theorem is referenced by: iccntr 23972 |
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