Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10890 | . . . 4 ⊢ -∞ ∈ ℝ* | |
2 | xrltnr 12711 | . . . 4 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ -∞ < -∞ |
4 | breq2 5057 | . . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
5 | 3, 4 | mtbiri 330 | . 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
6 | mnfle 12726 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
7 | xrleloe 12734 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) | |
8 | 1, 7 | mpan 690 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) |
9 | 6, 8 | mpbid 235 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ∨ -∞ = 𝐴)) |
10 | 9 | ord 864 | . . 3 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → -∞ = 𝐴)) |
11 | eqcom 2744 | . . 3 ⊢ (-∞ = 𝐴 ↔ 𝐴 = -∞) | |
12 | 10, 11 | syl6ib 254 | . 2 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → 𝐴 = -∞)) |
13 | 5, 12 | impbid2 229 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 847 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 -∞cmnf 10865 ℝ*cxr 10866 < clt 10867 ≤ cle 10868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 |
This theorem is referenced by: xlemnf 12757 xrrebnd 12758 ge0nemnf 12763 xlt2add 12850 xrsdsreclblem 20409 xblpnfps 23293 xblpnf 23294 supxrnemnf 30811 itg2addnclem 35565 supxrgelem 42549 supxrge 42550 nemnftgtmnft 42556 infxrbnd2 42581 |
Copyright terms: Public domain | W3C validator |