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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzidd2 | Structured version Visualization version GIF version |
Description: Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
uzidd2.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
uzidd2.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
uzidd2 | ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzidd2.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | 1 | uzidd 12586 | . 2 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | uzidd2.2 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrrdi 2850 | 1 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6427 ℤcz 12307 ℤ≥cuz 12570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-pre-lttri 10933 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-neg 11196 df-z 12308 df-uz 12571 |
This theorem is referenced by: uzn0d 42924 uzub 42930 climinf2lem 43206 limsupmnfuzlem 43226 limsupre3uzlem 43235 |
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