![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9540 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | 1 | zred 9378 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ‘cfv 5218 ℝcr 7813 ℤ≥cuz 9531 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-cnex 7905 ax-resscn 7906 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5881 df-neg 8134 df-z 9257 df-uz 9532 |
This theorem is referenced by: eluzelcn 9542 fzouzdisj 10183 eluzgtdifelfzo 10200 rebtwn2zlemstep 10256 m1modge3gt1 10374 bernneq3 10646 hashfzp1 10807 seq3coll 10825 sumsnf 11420 infssuzex 11953 infssuzledc 11954 isprm5 12145 dfphi2 12223 pclemub 12290 pockthg 12358 logbrec 14518 logbleb 14519 logbgcd1irr 14525 |
Copyright terms: Public domain | W3C validator |