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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 10631 | . 2 ⊢ 0 ∈ ℝ | |
2 | ltpnf 12503 | . 2 ⊢ (0 ∈ ℝ → 0 < +∞) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 0 < +∞ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 class class class wbr 5057 ℝcr 10524 0cc0 10525 +∞cpnf 10660 < clt 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-pnf 10665 df-xr 10667 df-ltxr 10668 |
This theorem is referenced by: xmulgt0 12664 reltxrnmnf 12723 hashneq0 13713 hashge2el2dif 13826 sgnpnf 14440 pnfnei 21756 0bdop 29697 xlt2addrd 30408 xnn0gt0 30420 xrge0mulc1cn 31083 pnfneige0 31093 lmxrge0 31094 mbfposadd 34820 ftc1anclem5 34852 fourierdlem111 42379 fouriersw 42393 |
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