Mathbox for Glauco Siliprandi |
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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 2907 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | 2 | biimpi 218 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | eluzelz 12256 | . 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 1536 ∈ wcel 2113 ‘cfv 6358 ℤcz 11984 ℤ≥cuz 12246 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-neg 10876 df-z 11985 df-uz 12247 |
This theorem is referenced by: eluzelz2d 41693 uzublem 41710 uzinico 41842 limsupubuzlem 41999 limsupmnfuzlem 42013 limsupre3uzlem 42022 limsupvaluz2 42025 supcnvlimsup 42027 xlimclim2lem 42126 climxlim2 42133 xlimliminflimsup 42149 smflimmpt 43091 smflimsuplem3 43103 smflimsuplem4 43104 smflimsuplem5 43105 smflimsuplem6 43106 smflimsuplem7 43107 smflimsuplem8 43108 smflimsupmpt 43110 smfliminflem 43111 smfliminfmpt 43113 |
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