Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mnfle | Structured version Visualization version GIF version |
Description: Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
mnfle | ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nltmnf 12527 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) | |
2 | mnfxr 10700 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | xrlenlt 10708 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) | |
4 | 2, 3 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) |
5 | 1, 4 | mpbird 259 | 1 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∈ wcel 2114 class class class wbr 5068 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: ngtmnft 12562 xrre2 12566 xleadd1a 12649 xlt2add 12656 xsubge0 12657 xlesubadd 12659 xlemul1a 12684 supxrmnf 12713 elioc2 12802 iccmax 12815 xrsdsreclblem 20593 leordtvallem2 21821 lecldbas 21829 tgioo 23406 xrtgioo 23416 ioombl 24168 ismbfd 24242 degltlem1 24668 ply1rem 24759 xrdifh 30505 tpr2rico 31157 itg2gt0cn 34949 hbtlem2 39731 supxrgelem 41612 supxrge 41613 suplesup 41614 xrlexaddrp 41627 infxr 41642 infleinf 41647 mnfled 41667 eliocre 41792 fouriersw 42523 |
Copyright terms: Public domain | W3C validator |