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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 2944 | . . 3 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
3 | nepnfltpnf.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | nltpnft 13202 | . . . 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 1540 ∈ wcel 2108 ≠ wne 2939 class class class wbr 5141 +∞cpnf 11288 ℝ*cxr 11290 < clt 11291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-pre-lttri 11225 ax-pre-lttrn 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 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 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 |
This theorem is referenced by: infrpge 45335 |
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