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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 | ⊢ (1 / 0) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-div 11376 | . . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
2 | riotaex 7131 | . . 3 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V | |
3 | 1, 2 | dmmpo 7794 | . 2 ⊢ dom / = (ℂ × (ℂ ∖ {0})) |
4 | eqid 2738 | . . 3 ⊢ 0 = 0 | |
5 | eldifsni 4678 | . . . . 5 ⊢ (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0) | |
6 | 5 | adantl 485 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0) |
7 | 6 | necon2bi 2964 | . . 3 ⊢ (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) |
8 | 4, 7 | ax-mp 5 | . 2 ⊢ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) |
9 | ndmovg 7347 | . 2 ⊢ ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅) | |
10 | 3, 8, 9 | mp2an 692 | 1 ⊢ (1 / 0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∖ cdif 3840 ∅c0 4211 {csn 4516 × cxp 5523 dom cdm 5525 ℩crio 7126 (class class class)co 7170 ℂcc 10613 0cc0 10615 1c1 10616 · cmul 10620 / cdiv 11375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-div 11376 |
This theorem is referenced by: (None) |
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