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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 | df-ee 28817 | . 2 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛))) | |
2 | 1 | mptrcl 7017 | 1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7423 ↑m cmap 8854 ℝcr 11153 1c1 11155 ℕcn 12259 ...cfz 13533 𝔼cee 28814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-xp 5687 df-rel 5688 df-cnv 5689 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fv 6561 df-ee 28817 |
This theorem is referenced by: eleei 28823 eedimeq 28824 brbtwn 28825 brcgr 28826 eleesub 28837 eleesubd 28838 axsegconlem1 28843 axsegconlem8 28850 axeuclidlem 28888 brsegle 35880 |
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