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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 10632 | . 2 ⊢ 0 ∈ ℝ | |
2 | mnflt 12506 | . 2 ⊢ (0 ∈ ℝ → -∞ < 0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -∞ < 0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 class class class wbr 5030 ℝcr 10525 0cc0 10526 -∞cmnf 10662 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 |
This theorem is referenced by: ge0gtmnf 12553 xsubge0 12642 sgnmnf 14446 leordtval2 21817 mnfnei 21826 ovolicopnf 24128 voliunlem3 24156 volsup 24160 volivth 24211 itg2seq 24346 itg2monolem2 24355 deg1lt0 24692 plypf1 24809 xrge00 30720 dvasin 35141 hbtlem5 40072 xrge0nemnfd 41964 xrpnf 42125 fourierdlem87 42835 fouriersw 42873 gsumge0cl 43010 sge0pr 43033 sge0ssre 43036 |
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