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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 26679 | . 2 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛))) | |
2 | 1 | mptrcl 6779 | 1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℝcr 10538 1c1 10540 ℕcn 11640 ...cfz 12895 𝔼cee 26676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fv 6365 df-ee 26679 |
This theorem is referenced by: eleei 26685 eedimeq 26686 brbtwn 26687 brcgr 26688 eleesub 26699 eleesubd 26700 axsegconlem1 26705 axsegconlem8 26712 axeuclidlem 26750 brsegle 33571 |
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