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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 7880 | . 2 ⊢ 0 ∈ ℝ | |
2 | mnflt 9696 | . 2 ⊢ (0 ∈ ℝ → -∞ < 0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -∞ < 0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 class class class wbr 3967 ℝcr 7733 0cc0 7734 -∞cmnf 7912 < clt 7914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-cnex 7825 ax-1re 7828 ax-addrcl 7831 ax-rnegex 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-xp 4594 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 |
This theorem is referenced by: ge0gtmnf 9733 xsubge0 9791 |
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