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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 4068 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → ¬ (𝔼‘𝑁) = ∅) | |
2 | ovex 6823 | . . . . 5 ⊢ (ℝ ↑𝑚 (1...𝑛)) ∈ V | |
3 | df-ee 25992 | . . . . 5 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛))) | |
4 | 2, 3 | dmmpti 6163 | . . . 4 ⊢ dom 𝔼 = ℕ |
5 | 4 | eleq2i 2842 | . . 3 ⊢ (𝑁 ∈ dom 𝔼 ↔ 𝑁 ∈ ℕ) |
6 | ndmfv 6359 | . . 3 ⊢ (¬ 𝑁 ∈ dom 𝔼 → (𝔼‘𝑁) = ∅) | |
7 | 5, 6 | sylnbir 320 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → (𝔼‘𝑁) = ∅) |
8 | 1, 7 | nsyl2 144 | 1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ∅c0 4063 dom cdm 5249 ‘cfv 6031 (class class class)co 6793 ↑𝑚 cmap 8009 ℝcr 10137 1c1 10139 ℕcn 11222 ...cfz 12533 𝔼cee 25989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fn 6034 df-fv 6039 df-ov 6796 df-ee 25992 |
This theorem is referenced by: eleei 25998 eedimeq 25999 brbtwn 26000 brcgr 26001 eleesub 26012 eleesubd 26013 axsegconlem1 26018 axsegconlem8 26025 axeuclidlem 26063 brsegle 32552 |
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