Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mnflt0 | GIF version |
Description: Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
mnflt0 | ⊢ -∞ < 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7119 | . 2 ⊢ 0 ∈ ℝ | |
2 | mnflt 8858 | . 2 ⊢ (0 ∈ ℝ → -∞ < 0) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ -∞ < 0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 0cc0 6981 -∞cmnf 7151 < clt 7153 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-cnex 7067 ax-1re 7070 ax-addrcl 7073 ax-rnegex 7085 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 |
This theorem is referenced by: ge0gtmnf 8890 |
Copyright terms: Public domain | W3C validator |