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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 11258 | . 2 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
2 | 1 | ssriv 3572 | 1 ⊢ ℤ ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3540 ℝcr 9814 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-neg 10148 df-z 11255 |
This theorem is referenced by: suprzcl 11333 zred 11358 suprfinzcl 11368 uzwo2 11628 infssuzle 11647 infssuzcl 11648 lbzbi 11652 suprzub 11655 uzwo3 11659 rpnnen1lem3 11692 rpnnen1lem5 11694 rpnnen1lem3OLD 11698 rpnnen1lem5OLD 11700 fzval2 12200 flval3 12478 uzsup 12524 expcan 12775 ltexp2 12776 seqcoll 13105 limsupgre 14060 rlimclim 14125 isercolllem1 14243 isercolllem2 14244 isercoll 14246 caurcvg 14255 caucvg 14257 summolem2a 14293 summolem2 14294 zsum 14296 fsumcvg3 14307 climfsum 14393 prodmolem2a 14503 prodmolem2 14504 zprod 14506 1arith 15469 pgpssslw 17852 gsumval3 18131 zntoslem 19724 zcld 22424 mbflimsup 23239 ig1pdvds 23740 aacjcl 23886 aalioulem3 23893 rzgrp 24104 qqhre 29392 ballotlemfc0 29881 ballotlemfcc 29882 ballotlemiex 29890 erdszelem4 30430 erdszelem8 30434 supfz 30866 inffz 30867 inffzOLD 30868 poimirlem31 32610 poimirlem32 32611 irrapxlem1 36404 monotuz 36524 monotoddzzfi 36525 rmyeq0 36538 rmyeq 36539 lermy 36540 fzisoeu 38455 fzssre 38470 uzfissfz 38483 ssuzfz 38506 ioodvbdlimc1lem2 38822 ioodvbdlimc2lem 38824 dvnprodlem1 38836 fourierdlem25 39025 fourierdlem37 39037 fourierdlem52 39051 fourierdlem64 39063 fourierdlem79 39078 etransclem48 39175 hoicvr 39438 |
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