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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 9185 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | 1 | zred 9025 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1448 ‘cfv 5059 ℝcr 7499 ℤ≥cuz 9176 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-cnex 7586 ax-resscn 7587 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-ov 5709 df-neg 7807 df-z 8907 df-uz 9177 |
This theorem is referenced by: eluzelcn 9187 fzouzdisj 9798 eluzgtdifelfzo 9815 rebtwn2zlemstep 9871 m1modge3gt1 9985 bernneq3 10255 hashfzp1 10411 seq3coll 10426 sumsnf 11017 infssuzex 11437 infssuzledc 11438 dfphi2 11688 |
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