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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 4295 | . . 3 β’ Β¬ π΅ β β | |
2 | pnfxr 11216 | . . . . . . . 8 β’ +β β β* | |
3 | xrltnr 13047 | . . . . . . . 8 β’ (+β β β* β Β¬ +β < +β) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 β’ Β¬ +β < +β |
5 | zre 12510 | . . . . . . . 8 β’ (+β β β€ β +β β β) | |
6 | ltpnf 13048 | . . . . . . . 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 488 | . . . . 5 β’ Β¬ (0 β β€ β§ +β β β€) |
10 | fzf 13435 | . . . . . . 7 β’ ...:(β€ Γ β€)βΆπ« β€ | |
11 | 10 | fdmi 6685 | . . . . . 6 β’ dom ... = (β€ Γ β€) |
12 | 11 | ndmov 7543 | . . . . 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 2830 | . . 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 397 = wceq 1542 β wcel 2107 β c0 4287 π« cpw 4565 class class class wbr 5110 Γ cxp 5636 (class class class)co 7362 βcr 11057 0cc0 11058 +βcpnf 11193 β*cxr 11195 < clt 11196 β€cz 12506 ...cfz 13431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-neg 11395 df-z 12507 df-fz 13432 |
This theorem is referenced by: (None) |
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