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| Mirrors > Home > MPE Home > Th. List > 0ltpnf | Structured version Visualization version GIF version | ||
| Description: Zero is less than plus infinity. (Contributed by David A. Wheeler, 8-Dec-2018.) |
| Ref | Expression |
|---|---|
| 0ltpnf | ⊢ 0 < +∞ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0re 11263 | . 2 ⊢ 0 ∈ ℝ | |
| 2 | ltpnf 13162 | . 2 ⊢ (0 ∈ ℝ → 0 < +∞) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 0 < +∞ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 0cc0 11155 +∞cpnf 11292 < clt 11295 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-pnf 11297 df-xr 11299 df-ltxr 11300 |
| This theorem is referenced by: xmulgt0 13325 reltxrnmnf 13384 hashneq0 14403 hashge2el2dif 14519 sgnpnf 15132 pnfnei 23228 0bdop 32012 xlt2addrd 32762 xnn0gt0 32773 xrge0mulc1cn 33940 pnfneige0 33950 lmxrge0 33951 mbfposadd 37674 ftc1anclem5 37704 fourierdlem111 46232 fouriersw 46246 |
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