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Mirrors > Home > MPE Home > Th. List > topnfbey | Structured version Visualization version GIF version |
Description: Nothing seems to be impossible to Prof. Lirpa. After years of intensive research, he managed to find a proof that when given a chance to reach infinity, one could indeed go beyond, thus giving formal soundness to Buzz Lightyear's motto "To infinity... and beyond!" (Contributed by Prof. Loof Lirpa, 1-Apr-2020.) (Revised by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
topnfbey | ⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4183 | . . 3 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | pnfxr 10494 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
3 | xrltnr 12331 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 ⊢ ¬ +∞ < +∞ |
5 | zre 11797 | . . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ) | |
6 | ltpnf 12332 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞) |
8 | 4, 7 | mto 189 | . . . . . 6 ⊢ ¬ +∞ ∈ ℤ |
9 | 8 | intnan 479 | . . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) |
10 | fzf 12712 | . . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
11 | 10 | fdmi 6354 | . . . . . 6 ⊢ dom ... = (ℤ × ℤ) |
12 | 11 | ndmov 7148 | . . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅) |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ (0...+∞) = ∅ |
14 | 13 | eleq2i 2857 | . . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅) |
15 | 1, 14 | mtbir 315 | . 2 ⊢ ¬ 𝐵 ∈ (0...+∞) |
16 | 15 | pm2.21i 117 | 1 ⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∅c0 4178 𝒫 cpw 4422 class class class wbr 4929 × cxp 5405 (class class class)co 6976 ℝcr 10334 0cc0 10335 +∞cpnf 10471 ℝ*cxr 10473 < clt 10474 ℤcz 11793 ...cfz 12708 |
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 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-pre-lttri 10409 ax-pre-lttrn 10410 |
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 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-1st 7501 df-2nd 7502 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-neg 10673 df-z 11794 df-fz 12709 |
This theorem is referenced by: (None) |
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