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 10692 | . . . 4 ⊢ -∞ ∈ ℝ* | |
2 | xrltnr 12508 | . . . 4 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ -∞ < -∞ |
4 | breq2 5062 | . . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
5 | 3, 4 | mtbiri 329 | . 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
6 | mnfle 12523 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
7 | xrleloe 12531 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) | |
8 | 1, 7 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) |
9 | 6, 8 | mpbid 234 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ∨ -∞ = 𝐴)) |
10 | 9 | ord 860 | . . 3 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → -∞ = 𝐴)) |
11 | eqcom 2828 | . . 3 ⊢ (-∞ = 𝐴 ↔ 𝐴 = -∞) | |
12 | 10, 11 | syl6ib 253 | . 2 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → 𝐴 = -∞)) |
13 | 5, 12 | impbid2 228 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 -∞cmnf 10667 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 |
This theorem is referenced by: xlemnf 12554 xrrebnd 12555 ge0nemnf 12560 xlt2add 12647 xrsdsreclblem 20585 xblpnfps 22999 xblpnf 23000 supxrnemnf 30487 itg2addnclem 34937 supxrgelem 41598 supxrge 41599 nemnftgtmnft 41605 infxrbnd2 41630 |
Copyright terms: Public domain | W3C validator |