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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 3920 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → ¬ (𝔼‘𝑁) = ∅) | |
2 | ovex 6678 | . . . . 5 ⊢ (ℝ ↑𝑚 (1...𝑛)) ∈ V | |
3 | df-ee 25771 | . . . . 5 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛))) | |
4 | 2, 3 | dmmpti 6023 | . . . 4 ⊢ dom 𝔼 = ℕ |
5 | 4 | eleq2i 2693 | . . 3 ⊢ (𝑁 ∈ dom 𝔼 ↔ 𝑁 ∈ ℕ) |
6 | ndmfv 6218 | . . 3 ⊢ (¬ 𝑁 ∈ dom 𝔼 → (𝔼‘𝑁) = ∅) | |
7 | 5, 6 | sylnbir 321 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → (𝔼‘𝑁) = ∅) |
8 | 1, 7 | nsyl2 142 | 1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∅c0 3915 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℝcr 9935 1c1 9937 ℕcn 11020 ...cfz 12326 𝔼cee 25768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ov 6653 df-ee 25771 |
This theorem is referenced by: eleei 25777 eedimeq 25778 brbtwn 25779 brcgr 25780 eleesub 25791 eleesubd 25792 axsegconlem1 25797 axsegconlem8 25804 axeuclidlem 25842 brsegle 32215 |
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