![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mnflt0 | Structured version Visualization version GIF version |
Description: Minus infinity is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
mnflt0 | ⊢ -∞ < 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11246 | . 2 ⊢ 0 ∈ ℝ | |
2 | mnflt 13135 | . 2 ⊢ (0 ∈ ℝ → -∞ < 0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -∞ < 0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 class class class wbr 5148 ℝcr 11137 0cc0 11138 -∞cmnf 11276 < clt 11278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-1cn 11196 ax-addrcl 11199 ax-rnegex 11209 ax-cnre 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5684 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 |
This theorem is referenced by: ge0gtmnf 13183 xsubge0 13272 sgnmnf 15074 leordtval2 23115 mnfnei 23124 ovolicopnf 25452 voliunlem3 25480 volsup 25484 volivth 25535 itg2seq 25671 itg2monolem2 25680 deg1lt0 26026 plypf1 26145 xrge00 32742 dvasin 37177 hbtlem5 42552 xrge0nemnfd 44714 xrpnf 44868 fourierdlem87 45581 fouriersw 45619 gsumge0cl 45759 sge0pr 45782 sge0ssre 45785 |
Copyright terms: Public domain | W3C validator |