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 9975 | . . . 4 ⊢ -∞ ∈ ℝ* | |
2 | xrltnr 11829 | . . . 4 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ -∞ < -∞ |
4 | breq2 4587 | . . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
5 | 3, 4 | mtbiri 316 | . 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
6 | mnfle 11845 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
7 | xrleloe 11853 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) | |
8 | 1, 7 | mpan 702 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴))) |
9 | 6, 8 | mpbid 221 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ∨ -∞ = 𝐴)) |
10 | 9 | ord 391 | . . 3 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → -∞ = 𝐴)) |
11 | eqcom 2617 | . . 3 ⊢ (-∞ = 𝐴 ↔ 𝐴 = -∞) | |
12 | 10, 11 | syl6ib 240 | . 2 ⊢ (𝐴 ∈ ℝ* → (¬ -∞ < 𝐴 → 𝐴 = -∞)) |
13 | 5, 12 | impbid2 215 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: xrrebnd 11873 ge0nemnf 11878 xlt2add 11962 xrsdsreclblem 19611 xblpnfps 22010 xblpnf 22011 xlemnf 28905 supxrnemnf 28924 itg2addnclem 32631 supxrgelem 38494 supxrge 38495 nemnftgtmnft 38501 infxrbnd2 38526 |
Copyright terms: Public domain | W3C validator |