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Mirrors > Home > MPE Home > Th. List > Mathboxes > eluzelz2d | 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 |
---|---|
eluzelz2d.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
eluzelz2d.2 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
Ref | Expression |
---|---|
eluzelz2d | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz2d.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
2 | eluzelz2d.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | eluzelz2 41552 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-neg 10861 df-z 11970 df-uz 12232 |
This theorem is referenced by: uzred 41593 limsupequzmpt2 41875 liminfequzmpt2 41948 xlimconst2 41992 smflimsuplem1 42971 smflimsuplem4 42974 smflimsuplem8 42978 smfliminflem 42981 |
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