Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eluzelz2 | Structured version Visualization version GIF version |
Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
eluzelz2.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
eluzelz2 | ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz2.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | eleq2i 2693 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | 2 | biimpi 206 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | eluzelz 11697 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 ℤcz 11377 ℤ≥cuz 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-neg 10269 df-z 11378 df-uz 11688 |
This theorem is referenced by: eluzelz2d 39640 uzublem 39657 uzinico 39787 limsupubuzlem 39944 limsupmnfuzlem 39958 limsupre3uzlem 39967 limsupvaluz2 39970 supcnvlimsup 39972 xlimclim2lem 40065 climxlim2 40072 smflimmpt 41016 smflimsuplem3 41028 smflimsuplem4 41029 smflimsuplem5 41030 smflimsuplem6 41031 smflimsuplem7 41032 smflimsuplem8 41033 smflimsupmpt 41035 smfliminflem 41036 smfliminfmpt 41038 |
Copyright terms: Public domain | W3C validator |