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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 4293 | . . 3 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | pnfxr 10683 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
3 | xrltnr 12502 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 ⊢ ¬ +∞ < +∞ |
5 | zre 11973 | . . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ) | |
6 | ltpnf 12503 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞) |
8 | 4, 7 | mto 198 | . . . . . 6 ⊢ ¬ +∞ ∈ ℤ |
9 | 8 | intnan 487 | . . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) |
10 | fzf 12884 | . . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
11 | 10 | fdmi 6517 | . . . . . 6 ⊢ dom ... = (ℤ × ℤ) |
12 | 11 | ndmov 7321 | . . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅) |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ (0...+∞) = ∅ |
14 | 13 | eleq2i 2901 | . . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅) |
15 | 1, 14 | mtbir 324 | . 2 ⊢ ¬ 𝐵 ∈ (0...+∞) |
16 | 15 | pm2.21i 119 | 1 ⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 𝒫 cpw 4535 class class class wbr 5057 × cxp 5546 (class class class)co 7145 ℝcr 10524 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ℤcz 11969 ...cfz 12880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-neg 10861 df-z 11970 df-fz 12881 |
This theorem is referenced by: (None) |
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