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Mirrors > Home > ILE Home > Th. List > zred | GIF version |
Description: An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
zred.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
Ref | Expression |
---|---|
zred | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssre 8252 | . 2 ⊢ ℤ ⊆ ℝ | |
2 | zred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
3 | 1, 2 | sseldi 2943 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ℝcr 6888 ℤcz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-neg 7185 df-z 8246 |
This theorem is referenced by: zcnd 8361 eluzelre 8483 eluzadd 8501 eluzsub 8502 uzm1 8503 z2ge 8739 fztri3or 8903 fznlem 8905 fzdisj 8916 fzpreddisj 8933 fznatpl1 8938 uzdisj 8955 fzm1 8962 fz0fzdiffz0 8987 elfzmlbm 8988 elfzmlbp 8990 difelfznle 8993 nn0disj 8995 elfzolt3 9013 fzonel 9016 fzouzdisj 9036 fzonmapblen 9043 fzoaddel 9048 elfzonelfzo 9086 qtri3or 9098 qbtwnzlemstep 9103 qbtwnzlemex 9105 qbtwnz 9106 rebtwn2zlemstep 9107 rebtwn2z 9109 qbtwnrelemcalc 9110 qbtwnre 9111 qfraclt1 9122 qfracge0 9123 flqge 9124 flid 9126 flqltnz 9129 flqwordi 9130 flqaddz 9139 flqmulnn0 9141 btwnzge0 9142 2tnp1ge0ge0 9143 flhalf 9144 flltdivnn0lt 9146 fldiv4p1lem1div2 9147 ceiqge 9151 ceiqm1l 9153 ceiqle 9155 flqleceil 9159 flqeqceilz 9160 intfracq 9162 modqval 9166 modqge0 9174 modqlt 9175 modqmulnn 9184 frec2uzlt2d 9190 frec2uzf1od 9192 expival 9257 resqrexlemdecn 9610 fzomaxdiflem 9708 nn0seqcvgd 9880 |
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