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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 4231 | . . 3 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | pnfxr 10852 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
3 | xrltnr 12676 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 ⊢ ¬ +∞ < +∞ |
5 | zre 12145 | . . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ) | |
6 | ltpnf 12677 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞) |
8 | 4, 7 | mto 200 | . . . . . 6 ⊢ ¬ +∞ ∈ ℤ |
9 | 8 | intnan 490 | . . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) |
10 | fzf 13064 | . . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
11 | 10 | fdmi 6535 | . . . . . 6 ⊢ dom ... = (ℤ × ℤ) |
12 | 11 | ndmov 7370 | . . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅) |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ (0...+∞) = ∅ |
14 | 13 | eleq2i 2822 | . . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅) |
15 | 1, 14 | mtbir 326 | . 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 399 = wceq 1543 ∈ wcel 2112 ∅c0 4223 𝒫 cpw 4499 class class class wbr 5039 × cxp 5534 (class class class)co 7191 ℝcr 10693 0cc0 10694 +∞cpnf 10829 ℝ*cxr 10831 < clt 10832 ℤcz 12141 ...cfz 13060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-pre-lttri 10768 ax-pre-lttrn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-neg 11030 df-z 12142 df-fz 13061 |
This theorem is referenced by: (None) |
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