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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 10330 | . 2 ⊢ 0 ∈ ℝ | |
2 | mnflt 12204 | . 2 ⊢ (0 ∈ ℝ → -∞ < 0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -∞ < 0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 class class class wbr 4843 ℝcr 10223 0cc0 10224 -∞cmnf 10361 < clt 10363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-1cn 10282 ax-addrcl 10285 ax-rnegex 10295 ax-cnre 10297 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-xp 5318 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 |
This theorem is referenced by: ge0gtmnf 12252 xsubge0 12340 xrge0neqmnfOLD 12527 sgnmnf 14176 leordtval2 21345 mnfnei 21354 ovolicopnf 23632 voliunlem3 23660 volsup 23664 volivth 23715 itg2seq 23850 itg2monolem2 23859 deg1lt0 24192 plypf1 24309 xrge00 30202 dvasin 33984 hbtlem5 38483 xrge0nemnfd 40292 xrpnf 40459 fourierdlem87 41153 fouriersw 41191 gsumge0cl 41331 sge0pr 41354 sge0ssre 41357 |
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