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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pnfged | Structured version Visualization version GIF version |
Description: Plus infinity is an upper bound for extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
pnfged.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Ref | Expression |
---|---|
pnfged | ⊢ (𝜑 → 𝐴 ≤ +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfged.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | pnfge 13107 | . 2 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≤ +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5138 +∞cpnf 11242 ℝ*cxr 11244 ≤ cle 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-xp 5672 df-cnv 5674 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 |
This theorem is referenced by: xlimpnfvlem2 45038 xlimliminflimsup 45063 pimgtpnf2f 45906 |
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