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Mirrors > Home > MPE Home > Th. List > ngtmnft | Structured version Visualization version GIF version |
Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
ngtmnft | ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 11268 | . . . 4 ⊢ -∞ ∈ ℝ* | |
2 | xrltnr 13096 | . . . 4 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ -∞ < -∞ |
4 | breq2 5152 | . . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
5 | 3, 4 | mtbiri 327 | . 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
6 | mnfle 13111 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
7 | xrleloe 13120 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) | |
8 | 1, 7 | mpan 689 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) |
9 | 6, 8 | mpbid 231 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ∨ -∞ = 𝐴)) |
10 | 9 | ord 863 | . . 3 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → -∞ = 𝐴)) |
11 | eqcom 2740 | . . 3 ⊢ (-∞ = 𝐴 ↔ 𝐴 = -∞) | |
12 | 10, 11 | imbitrdi 250 | . 2 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → 𝐴 = -∞)) |
13 | 5, 12 | impbid2 225 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5148 -∞cmnf 11243 ℝ*cxr 11244 < clt 11245 ≤ cle 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-pre-lttri 11181 ax-pre-lttrn 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 |
This theorem is referenced by: xlemnf 13143 xrrebnd 13144 ge0nemnf 13149 xlt2add 13236 xrsdsreclblem 20984 xblpnfps 23893 xblpnf 23894 supxrnemnf 31969 itg2addnclem 36528 supxrgelem 44034 supxrge 44035 nemnftgtmnft 44041 infxrbnd2 44066 |
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