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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0cnf | Structured version Visualization version GIF version |
Description: The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
0cnf | ⊢ ∅ ∈ ({∅} Cn {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 6772 | . 2 ⊢ ∅:∅⟶∅ | |
2 | cnv0 6139 | . . . . . 6 ⊢ ◡∅ = ∅ | |
3 | 2 | imaeq1i 6054 | . . . . 5 ⊢ (◡∅ “ 𝑥) = (∅ “ 𝑥) |
4 | 0ima 6075 | . . . . 5 ⊢ (∅ “ 𝑥) = ∅ | |
5 | 3, 4 | eqtri 2755 | . . . 4 ⊢ (◡∅ “ 𝑥) = ∅ |
6 | 0ex 5301 | . . . . 5 ⊢ ∅ ∈ V | |
7 | 6 | snid 4660 | . . . 4 ⊢ ∅ ∈ {∅} |
8 | 5, 7 | eqeltri 2824 | . . 3 ⊢ (◡∅ “ 𝑥) ∈ {∅} |
9 | 8 | rgenw 3060 | . 2 ⊢ ∀𝑥 ∈ {∅} (◡∅ “ 𝑥) ∈ {∅} |
10 | sn0topon 22888 | . . 3 ⊢ {∅} ∈ (TopOn‘∅) | |
11 | iscn 23126 | . . 3 ⊢ (({∅} ∈ (TopOn‘∅) ∧ {∅} ∈ (TopOn‘∅)) → (∅ ∈ ({∅} Cn {∅}) ↔ (∅:∅⟶∅ ∧ ∀𝑥 ∈ {∅} (◡∅ “ 𝑥) ∈ {∅}))) | |
12 | 10, 10, 11 | mp2an 691 | . 2 ⊢ (∅ ∈ ({∅} Cn {∅}) ↔ (∅:∅⟶∅ ∧ ∀𝑥 ∈ {∅} (◡∅ “ 𝑥) ∈ {∅})) |
13 | 1, 9, 12 | mpbir2an 710 | 1 ⊢ ∅ ∈ ({∅} Cn {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ∀wral 3056 ∅c0 4318 {csn 4624 ◡ccnv 5671 “ cima 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 TopOnctopon 22799 Cn ccn 23115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8838 df-top 22783 df-topon 22800 df-cn 23118 |
This theorem is referenced by: cncfiooicc 45205 |
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