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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nepnfltpnf | Structured version Visualization version GIF version |
Description: An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
nepnfltpnf.1 | ⊢ (𝜑 → 𝐴 ≠ +∞) |
nepnfltpnf.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Ref | Expression |
---|---|
nepnfltpnf | ⊢ (𝜑 → 𝐴 < +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nepnfltpnf.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ +∞) | |
2 | 1 | neneqd 2941 | . . 3 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
3 | nepnfltpnf.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | nltpnft 13196 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
6 | 2, 5 | mtbid 324 | . 2 ⊢ (𝜑 → ¬ ¬ 𝐴 < +∞) |
7 | notnotb 315 | . 2 ⊢ (𝐴 < +∞ ↔ ¬ ¬ 𝐴 < +∞) | |
8 | 6, 7 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 < +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 class class class wbr 5149 +∞cpnf 11283 ℝ*cxr 11285 < clt 11286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 |
This theorem is referenced by: infrpge 45251 |
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