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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 10564 | . . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
2 | riotaex 6515 | . . 3 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V | |
3 | 1, 2 | dmmpt2 7129 | . 2 ⊢ dom / = (ℂ × (ℂ ∖ {0})) |
4 | eqid 2610 | . . 3 ⊢ 0 = 0 | |
5 | eldifsni 4261 | . . . . 5 ⊢ (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0) | |
6 | 5 | adantl 481 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0) |
7 | 6 | necon2bi 2812 | . . 3 ⊢ (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) |
8 | 4, 7 | ax-mp 5 | . 2 ⊢ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) |
9 | ndmovg 6715 | . 2 ⊢ ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅) | |
10 | 3, 8, 9 | mp2an 704 | 1 ⊢ (1 / 0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ∅c0 3874 {csn 4125 × cxp 5036 dom cdm 5038 ℩crio 6510 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 / cdiv 10563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-div 10564 |
This theorem is referenced by: (None) |
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