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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 9466 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | 1 | zred 9304 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ‘cfv 5182 ℝcr 7743 ℤ≥cuz 9457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-neg 8063 df-z 9183 df-uz 9458 |
This theorem is referenced by: eluzelcn 9468 fzouzdisj 10105 eluzgtdifelfzo 10122 rebtwn2zlemstep 10178 m1modge3gt1 10296 bernneq3 10566 hashfzp1 10726 seq3coll 10741 sumsnf 11336 infssuzex 11867 infssuzledc 11868 isprm5 12053 dfphi2 12131 pclemub 12198 pockthg 12266 logbrec 13425 logbleb 13426 logbgcd1irr 13432 |
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