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Mirrors > Home > ILE Home > Th. List > eluzelre | GIF version |
Description: A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.) |
Ref | Expression |
---|---|
eluzelre | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9591 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | 1 | zred 9429 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 ‘cfv 5246 ℝcr 7861 ℤ≥cuz 9582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-cnex 7953 ax-resscn 7954 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-fv 5254 df-ov 5913 df-neg 8183 df-z 9308 df-uz 9583 |
This theorem is referenced by: eluzelcn 9593 fzouzdisj 10237 eluzgtdifelfzo 10254 rebtwn2zlemstep 10311 fldiv4lem1div2uz2 10365 m1modge3gt1 10432 bernneq3 10720 hashfzp1 10882 seq3coll 10900 sumsnf 11539 infssuzex 12073 infssuzledc 12074 isprm5 12267 dfphi2 12345 pclemub 12412 pockthg 12482 gsumfzval 12961 logbrec 15034 logbleb 15035 logbgcd1irr 15041 gausslemma2dlem4 15122 |
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