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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 11912 | . . 3 โข / = (๐ฅ โ โ, ๐ฆ โ (โ โ {0}) โฆ (โฉ๐ง โ โ (๐ฆ ยท ๐ง) = ๐ฅ)) | |
2 | riotaex 7386 | . . 3 โข (โฉ๐ง โ โ (๐ฆ ยท ๐ง) = ๐ฅ) โ V | |
3 | 1, 2 | dmmpo 8083 | . 2 โข dom / = (โ ร (โ โ {0})) |
4 | eqid 2728 | . . 3 โข 0 = 0 | |
5 | eldifsni 4798 | . . . . 5 โข (0 โ (โ โ {0}) โ 0 โ 0) | |
6 | 5 | adantl 480 | . . . 4 โข ((1 โ โ โง 0 โ (โ โ {0})) โ 0 โ 0) |
7 | 6 | necon2bi 2968 | . . 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 7611 | . 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 690 | 1 โข (1 / 0) = โ |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 โง wa 394 = wceq 1533 โ wcel 2098 โ wne 2937 โ cdif 3946 โ c0 4326 {csn 4632 ร cxp 5680 dom cdm 5682 โฉcrio 7381 (class class class)co 7426 โcc 11146 0cc0 11148 1c1 11149 ยท cmul 11153 / cdiv 11911 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-div 11912 |
This theorem is referenced by: (None) |
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