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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 11275 | . . . 4 ⊢ -∞ ∈ ℝ* | |
2 | xrltnr 13103 | . . . 4 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ -∞ < -∞ |
4 | breq2 5151 | . . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
5 | 3, 4 | mtbiri 326 | . 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
6 | mnfle 13118 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
7 | xrleloe 13127 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) | |
8 | 1, 7 | mpan 686 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) |
9 | 6, 8 | mpbid 231 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ∨ -∞ = 𝐴)) |
10 | 9 | ord 860 | . . 3 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → -∞ = 𝐴)) |
11 | eqcom 2737 | . . 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 843 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 -∞cmnf 11250 ℝ*cxr 11251 < clt 11252 ≤ cle 11253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 |
This theorem is referenced by: xlemnf 13150 xrrebnd 13151 ge0nemnf 13156 xlt2add 13243 xrsdsreclblem 21191 xblpnfps 24121 xblpnf 24122 supxrnemnf 32248 itg2addnclem 36842 supxrgelem 44345 supxrge 44346 nemnftgtmnft 44352 infxrbnd2 44377 |
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