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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 | β’ (π΅ β (0...+β) β +β < π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4325 | . . 3 β’ Β¬ π΅ β β | |
2 | pnfxr 11272 | . . . . . . . 8 β’ +β β β* | |
3 | xrltnr 13105 | . . . . . . . 8 β’ (+β β β* β Β¬ +β < +β) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 β’ Β¬ +β < +β |
5 | zre 12566 | . . . . . . . 8 β’ (+β β β€ β +β β β) | |
6 | ltpnf 13106 | . . . . . . . 8 β’ (+β β β β +β < +β) | |
7 | 5, 6 | syl 17 | . . . . . . 7 β’ (+β β β€ β +β < +β) |
8 | 4, 7 | mto 196 | . . . . . 6 β’ Β¬ +β β β€ |
9 | 8 | intnan 486 | . . . . 5 β’ Β¬ (0 β β€ β§ +β β β€) |
10 | fzf 13494 | . . . . . . 7 β’ ...:(β€ Γ β€)βΆπ« β€ | |
11 | 10 | fdmi 6723 | . . . . . 6 β’ dom ... = (β€ Γ β€) |
12 | 11 | ndmov 7588 | . . . . 5 β’ (Β¬ (0 β β€ β§ +β β β€) β (0...+β) = β ) |
13 | 9, 12 | ax-mp 5 | . . . 4 β’ (0...+β) = β |
14 | 13 | eleq2i 2819 | . . 3 β’ (π΅ β (0...+β) β π΅ β β ) |
15 | 1, 14 | mtbir 323 | . 2 β’ Β¬ π΅ β (0...+β) |
16 | 15 | pm2.21i 119 | 1 β’ (π΅ β (0...+β) β +β < π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β c0 4317 π« cpw 4597 class class class wbr 5141 Γ cxp 5667 (class class class)co 7405 βcr 11111 0cc0 11112 +βcpnf 11249 β*cxr 11251 < clt 11252 β€cz 12562 ...cfz 13490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-neg 11451 df-z 12563 df-fz 13491 |
This theorem is referenced by: (None) |
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