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Mirrors > Home > MPE Home > Th. List > eleenn | Structured version Visualization version GIF version |
Description: If 𝐴 is in (𝔼‘𝑁), then 𝑁 is a natural. (Contributed by Scott Fenton, 1-Jul-2013.) |
Ref | Expression |
---|---|
eleenn | ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4148 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → ¬ (𝔼‘𝑁) = ∅) | |
2 | ovex 6954 | . . . . 5 ⊢ (ℝ ↑𝑚 (1...𝑛)) ∈ V | |
3 | df-ee 26240 | . . . . 5 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛))) | |
4 | 2, 3 | dmmpti 6269 | . . . 4 ⊢ dom 𝔼 = ℕ |
5 | 4 | eleq2i 2851 | . . 3 ⊢ (𝑁 ∈ dom 𝔼 ↔ 𝑁 ∈ ℕ) |
6 | ndmfv 6476 | . . 3 ⊢ (¬ 𝑁 ∈ dom 𝔼 → (𝔼‘𝑁) = ∅) | |
7 | 5, 6 | sylnbir 323 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → (𝔼‘𝑁) = ∅) |
8 | 1, 7 | nsyl2 145 | 1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∅c0 4141 dom cdm 5355 ‘cfv 6135 (class class class)co 6922 ↑𝑚 cmap 8140 ℝcr 10271 1c1 10273 ℕcn 11374 ...cfz 12643 𝔼cee 26237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 df-ov 6925 df-ee 26240 |
This theorem is referenced by: eleei 26246 eedimeq 26247 brbtwn 26248 brcgr 26249 eleesub 26260 eleesubd 26261 axsegconlem1 26266 axsegconlem8 26273 axeuclidlem 26311 brsegle 32804 |
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