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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 11876 | . . 3 โข / = (๐ฅ โ โ, ๐ฆ โ (โ โ {0}) โฆ (โฉ๐ง โ โ (๐ฆ ยท ๐ง) = ๐ฅ)) | |
2 | riotaex 7365 | . . 3 โข (โฉ๐ง โ โ (๐ฆ ยท ๐ง) = ๐ฅ) โ V | |
3 | 1, 2 | dmmpo 8056 | . 2 โข dom / = (โ ร (โ โ {0})) |
4 | eqid 2726 | . . 3 โข 0 = 0 | |
5 | eldifsni 4788 | . . . . 5 โข (0 โ (โ โ {0}) โ 0 โ 0) | |
6 | 5 | adantl 481 | . . . 4 โข ((1 โ โ โง 0 โ (โ โ {0})) โ 0 โ 0) |
7 | 6 | necon2bi 2965 | . . 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 7587 | . 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 689 | 1 โข (1 / 0) = โ |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 โง wa 395 = wceq 1533 โ wcel 2098 โ wne 2934 โ cdif 3940 โ c0 4317 {csn 4623 ร cxp 5667 dom cdm 5669 โฉcrio 7360 (class class class)co 7405 โcc 11110 0cc0 11112 1c1 11113 ยท cmul 11117 / cdiv 11875 |
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-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-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-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-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-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-div 11876 |
This theorem is referenced by: (None) |
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