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 12519 | . 2 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≤ +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 class class class wbr 5059 +∞cpnf 10665 ℝ*cxr 10667 ≤ cle 10669 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-cnv 5556 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 |
This theorem is referenced by: xlimpnfvlem2 42192 xlimliminflimsup 42217 |
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