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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzssd3 | Structured version Visualization version GIF version |
Description: Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
uzssd3.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
uzssd3 | ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzssd3.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | id 22 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ 𝑍) | |
3 | 1, 2 | uzssd2 45317 | 1 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ⊆ wss 3963 ‘cfv 6558 ℤ≥cuz 12869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-neg 11486 df-z 12605 df-uz 12870 |
This theorem is referenced by: limsupvaluzmpt 45623 limsupvaluz2 45644 supcnvlimsupmpt 45647 climxlim2 45752 meaiuninc3v 46390 smflimmpt 46716 |
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