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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 2966 | . . 3 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
3 | nepnfltpnf.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | nltpnft 12368 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
6 | 2, 5 | mtbid 316 | . 2 ⊢ (𝜑 → ¬ ¬ 𝐴 < +∞) |
7 | notnotb 307 | . 2 ⊢ (𝐴 < +∞ ↔ ¬ ¬ 𝐴 < +∞) | |
8 | 6, 7 | sylibr 226 | 1 ⊢ (𝜑 → 𝐴 < +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 class class class wbr 4923 +∞cpnf 10465 ℝ*cxr 10467 < clt 10468 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-pre-lttri 10403 ax-pre-lttrn 10404 |
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 2016 df-mo 2547 df-eu 2584 df-clab 2753 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 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 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 5306 df-po 5320 df-so 5321 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 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 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 |
This theorem is referenced by: infrpge 41048 |
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