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| Mirrors > Home > MPE Home > Th. List > zssre | Structured version Visualization version GIF version | ||
| Description: The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
| Ref | Expression |
|---|---|
| zssre | ⊢ ℤ ⊆ ℝ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zre 12617 | . 2 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
| 2 | 1 | ssriv 3987 | 1 ⊢ ℤ ⊆ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 ℝcr 11154 ℤcz 12613 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 |
| This theorem is referenced by: suprzcl 12698 zred 12722 suprfinzcl 12732 uzssre 12900 uzwo2 12954 infssuzle 12973 infssuzcl 12974 lbzbi 12978 suprzub 12981 uzwo3 12985 rpnnen1lem3 13021 rpnnen1lem5 13023 fzval2 13550 flval3 13855 uzsup 13903 expcan 14209 ltexp2 14210 seqcoll 14503 limsupgre 15517 rlimclim 15582 isercolllem1 15701 isercolllem2 15702 isercoll 15704 caurcvg 15713 caucvg 15715 summolem2a 15751 summolem2 15752 zsum 15754 fsumcvg3 15765 climfsum 15856 prodmolem2a 15970 prodmolem2 15971 zprod 15973 1arith 16965 pgpssslw 19632 gsumval3 19925 zntoslem 21575 rzgrp 21641 zcld 24835 mbflimsup 25701 ig1pdvds 26219 aacjcl 26369 aalioulem3 26376 uzssico 32786 qqhre 34021 ballotlemfc0 34495 ballotlemfcc 34496 ballotlemiex 34504 erdszelem4 35199 erdszelem8 35203 supfz 35729 inffz 35730 poimirlem31 37658 poimirlem32 37659 irrapxlem1 42833 monotuz 42953 monotoddzzfi 42954 rmyeq0 42965 rmyeq 42966 lermy 42967 fzisoeu 45312 fzssre 45326 uzfissfz 45337 ssuzfz 45360 zssxr 45408 uzssre2 45418 uzred 45454 uzinico 45573 ioodvbdlimc1lem2 45947 ioodvbdlimc2lem 45949 fourierdlem25 46147 fourierdlem37 46159 fourierdlem52 46173 fourierdlem64 46185 fourierdlem79 46200 etransclem48 46297 hoicvr 46563 |
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