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| 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 6788 | . 2 ⊢ ∅:∅⟶∅ | |
| 2 | cnv0 6159 | . . . . . 6 ⊢ ◡∅ = ∅ | |
| 3 | 2 | imaeq1i 6074 | . . . . 5 ⊢ (◡∅ “ 𝑥) = (∅ “ 𝑥) | 
| 4 | 0ima 6095 | . . . . 5 ⊢ (∅ “ 𝑥) = ∅ | |
| 5 | 3, 4 | eqtri 2764 | . . . 4 ⊢ (◡∅ “ 𝑥) = ∅ | 
| 6 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
| 7 | 6 | snid 4661 | . . . 4 ⊢ ∅ ∈ {∅} | 
| 8 | 5, 7 | eqeltri 2836 | . . 3 ⊢ (◡∅ “ 𝑥) ∈ {∅} | 
| 9 | 8 | rgenw 3064 | . 2 ⊢ ∀𝑥 ∈ {∅} (◡∅ “ 𝑥) ∈ {∅} | 
| 10 | sn0topon 23006 | . . 3 ⊢ {∅} ∈ (TopOn‘∅) | |
| 11 | iscn 23244 | . . 3 ⊢ (({∅} ∈ (TopOn‘∅) ∧ {∅} ∈ (TopOn‘∅)) → (∅ ∈ ({∅} Cn {∅}) ↔ (∅:∅⟶∅ ∧ ∀𝑥 ∈ {∅} (◡∅ “ 𝑥) ∈ {∅}))) | |
| 12 | 10, 10, 11 | mp2an 692 | . 2 ⊢ (∅ ∈ ({∅} Cn {∅}) ↔ (∅:∅⟶∅ ∧ ∀𝑥 ∈ {∅} (◡∅ “ 𝑥) ∈ {∅})) | 
| 13 | 1, 9, 12 | mpbir2an 711 | 1 ⊢ ∅ ∈ ({∅} Cn {∅}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ∅c0 4332 {csn 4625 ◡ccnv 5683 “ cima 5687 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 TopOnctopon 22917 Cn ccn 23233 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-top 22901 df-topon 22918 df-cn 23236 | 
| This theorem is referenced by: cncfiooicc 45914 | 
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