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Mirrors > Home > MPE Home > Th. List > 1div0apr | Structured version Visualization version GIF version |
Description: Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1div0apr | โข (1 / 0) = โ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-div 11820 | . . 3 โข / = (๐ฅ โ โ, ๐ฆ โ (โ โ {0}) โฆ (โฉ๐ง โ โ (๐ฆ ยท ๐ง) = ๐ฅ)) | |
2 | riotaex 7322 | . . 3 โข (โฉ๐ง โ โ (๐ฆ ยท ๐ง) = ๐ฅ) โ V | |
3 | 1, 2 | dmmpo 8008 | . 2 โข dom / = (โ ร (โ โ {0})) |
4 | eqid 2737 | . . 3 โข 0 = 0 | |
5 | eldifsni 4755 | . . . . 5 โข (0 โ (โ โ {0}) โ 0 โ 0) | |
6 | 5 | adantl 483 | . . . 4 โข ((1 โ โ โง 0 โ (โ โ {0})) โ 0 โ 0) |
7 | 6 | necon2bi 2975 | . . 3 โข (0 = 0 โ ยฌ (1 โ โ โง 0 โ (โ โ {0}))) |
8 | 4, 7 | ax-mp 5 | . 2 โข ยฌ (1 โ โ โง 0 โ (โ โ {0})) |
9 | ndmovg 7542 | . 2 โข ((dom / = (โ ร (โ โ {0})) โง ยฌ (1 โ โ โง 0 โ (โ โ {0}))) โ (1 / 0) = โ ) | |
10 | 3, 8, 9 | mp2an 691 | 1 โข (1 / 0) = โ |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 โง wa 397 = wceq 1542 โ wcel 2107 โ wne 2944 โ cdif 3912 โ c0 4287 {csn 4591 ร cxp 5636 dom cdm 5638 โฉcrio 7317 (class class class)co 7362 โcc 11056 0cc0 11058 1c1 11059 ยท cmul 11063 / cdiv 11819 |
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-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-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-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-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-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-div 11820 |
This theorem is referenced by: (None) |
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