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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nemnftgtmnft | Structured version Visualization version GIF version |
Description: An extended real that is not minus infinity, is larger than minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
nemnftgtmnft | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞) | |
2 | 1 | neneqd 2966 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → ¬ 𝐴 = -∞) |
3 | ngtmnft 12369 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) | |
4 | 3 | adantr 473 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
5 | 2, 4 | mtbid 316 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → ¬ ¬ -∞ < 𝐴) |
6 | notnotb 307 | . 2 ⊢ (-∞ < 𝐴 ↔ ¬ ¬ -∞ < 𝐴) | |
7 | 5, 6 | sylibr 226 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 class class class wbr 4923 -∞cmnf 10464 ℝ*cxr 10465 < clt 10466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-pre-lttri 10401 ax-pre-lttrn 10402 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 |
This theorem is referenced by: xrlexaddrp 40995 |
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