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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 4294 | . . 3 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | pnfxr 10687 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
3 | xrltnr 12506 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 ⊢ ¬ +∞ < +∞ |
5 | zre 11977 | . . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ) | |
6 | ltpnf 12507 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞) |
8 | 4, 7 | mto 199 | . . . . . 6 ⊢ ¬ +∞ ∈ ℤ |
9 | 8 | intnan 489 | . . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) |
10 | fzf 12888 | . . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
11 | 10 | fdmi 6517 | . . . . . 6 ⊢ dom ... = (ℤ × ℤ) |
12 | 11 | ndmov 7324 | . . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅) |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ (0...+∞) = ∅ |
14 | 13 | eleq2i 2902 | . . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅) |
15 | 1, 14 | mtbir 325 | . 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 398 = wceq 1531 ∈ wcel 2108 ∅c0 4289 𝒫 cpw 4537 class class class wbr 5057 × cxp 5546 (class class class)co 7148 ℝcr 10528 0cc0 10529 +∞cpnf 10664 ℝ*cxr 10666 < clt 10667 ℤcz 11973 ...cfz 12884 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 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 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-neg 10865 df-z 11974 df-fz 12885 |
This theorem is referenced by: (None) |
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