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Mirrors > Home > MPE Home > Th. List > avril1 | Structured version Visualization version GIF version |
Description: Poisson d'Avril's
Theorem. This theorem is noted for its
Selbstdokumentieren property, which means, literally,
"self-documenting" and recalls the principle of quidquid
german dictum
sit, altum viditur, often used in set theory. Starting with the
seemingly simple yet profound fact that any object 𝑥 equals
itself
(proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we
demonstrate that the power set of the real numbers, as a relation on the
value of the imaginary unit, does not conjoin with an empty relation on
the product of the additive and multiplicative identity elements,
leading to this startling conclusion that has left even seasoned
professional mathematicians scratching their heads. (Contributed by
Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.)
(New usage is discouraged.)
A reply to skeptics can be found at mmnotes.txt, under the 1-Apr-2006 entry. |
Ref | Expression |
---|---|
avril1 | ⊢ ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2019 | . . . . . . . 8 ⊢ 𝑥 = 𝑥 | |
2 | dfnul2 4295 | . . . . . . . . . 10 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
3 | 2 | abeq2i 2950 | . . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥) |
4 | 3 | con2bii 360 | . . . . . . . 8 ⊢ (𝑥 = 𝑥 ↔ ¬ 𝑥 ∈ ∅) |
5 | 1, 4 | mpbi 232 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ |
6 | eleq1 2902 | . . . . . . 7 ⊢ (𝑥 = 〈𝐹, 0〉 → (𝑥 ∈ ∅ ↔ 〈𝐹, 0〉 ∈ ∅)) | |
7 | 5, 6 | mtbii 328 | . . . . . 6 ⊢ (𝑥 = 〈𝐹, 0〉 → ¬ 〈𝐹, 0〉 ∈ ∅) |
8 | 7 | vtocleg 3583 | . . . . 5 ⊢ (〈𝐹, 0〉 ∈ V → ¬ 〈𝐹, 0〉 ∈ ∅) |
9 | elex 3514 | . . . . . 6 ⊢ (〈𝐹, 0〉 ∈ ∅ → 〈𝐹, 0〉 ∈ V) | |
10 | 9 | con3i 157 | . . . . 5 ⊢ (¬ 〈𝐹, 0〉 ∈ V → ¬ 〈𝐹, 0〉 ∈ ∅) |
11 | 8, 10 | pm2.61i 184 | . . . 4 ⊢ ¬ 〈𝐹, 0〉 ∈ ∅ |
12 | df-br 5069 | . . . . 5 ⊢ (𝐹∅(0 · 1) ↔ 〈𝐹, (0 · 1)〉 ∈ ∅) | |
13 | 0cn 10635 | . . . . . . . 8 ⊢ 0 ∈ ℂ | |
14 | 13 | mulid1i 10647 | . . . . . . 7 ⊢ (0 · 1) = 0 |
15 | 14 | opeq2i 4809 | . . . . . 6 ⊢ 〈𝐹, (0 · 1)〉 = 〈𝐹, 0〉 |
16 | 15 | eleq1i 2905 | . . . . 5 ⊢ (〈𝐹, (0 · 1)〉 ∈ ∅ ↔ 〈𝐹, 0〉 ∈ ∅) |
17 | 12, 16 | bitri 277 | . . . 4 ⊢ (𝐹∅(0 · 1) ↔ 〈𝐹, 0〉 ∈ ∅) |
18 | 11, 17 | mtbir 325 | . . 3 ⊢ ¬ 𝐹∅(0 · 1) |
19 | 18 | intnan 489 | . 2 ⊢ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1〈0R, 1R〉𝑦) ∧ 𝐹∅(0 · 1)) |
20 | df-i 10548 | . . . . . . . 8 ⊢ i = 〈0R, 1R〉 | |
21 | 20 | fveq1i 6673 | . . . . . . 7 ⊢ (i‘1) = (〈0R, 1R〉‘1) |
22 | df-fv 6365 | . . . . . . 7 ⊢ (〈0R, 1R〉‘1) = (℩𝑦1〈0R, 1R〉𝑦) | |
23 | 21, 22 | eqtri 2846 | . . . . . 6 ⊢ (i‘1) = (℩𝑦1〈0R, 1R〉𝑦) |
24 | 23 | breq2i 5076 | . . . . 5 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 ℝ(℩𝑦1〈0R, 1R〉𝑦)) |
25 | df-r 10549 | . . . . . . 7 ⊢ ℝ = (R × {0R}) | |
26 | sseq2 3995 | . . . . . . . . 9 ⊢ (ℝ = (R × {0R}) → (𝑧 ⊆ ℝ ↔ 𝑧 ⊆ (R × {0R}))) | |
27 | 26 | abbidv 2887 | . . . . . . . 8 ⊢ (ℝ = (R × {0R}) → {𝑧 ∣ 𝑧 ⊆ ℝ} = {𝑧 ∣ 𝑧 ⊆ (R × {0R})}) |
28 | df-pw 4543 | . . . . . . . 8 ⊢ 𝒫 ℝ = {𝑧 ∣ 𝑧 ⊆ ℝ} | |
29 | df-pw 4543 | . . . . . . . 8 ⊢ 𝒫 (R × {0R}) = {𝑧 ∣ 𝑧 ⊆ (R × {0R})} | |
30 | 27, 28, 29 | 3eqtr4g 2883 | . . . . . . 7 ⊢ (ℝ = (R × {0R}) → 𝒫 ℝ = 𝒫 (R × {0R})) |
31 | 25, 30 | ax-mp 5 | . . . . . 6 ⊢ 𝒫 ℝ = 𝒫 (R × {0R}) |
32 | 31 | breqi 5074 | . . . . 5 ⊢ (𝐴𝒫 ℝ(℩𝑦1〈0R, 1R〉𝑦) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1〈0R, 1R〉𝑦)) |
33 | 24, 32 | bitri 277 | . . . 4 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1〈0R, 1R〉𝑦)) |
34 | 33 | anbi1i 625 | . . 3 ⊢ ((𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ (𝐴𝒫 (R × {0R})(℩𝑦1〈0R, 1R〉𝑦) ∧ 𝐹∅(0 · 1))) |
35 | 34 | notbii 322 | . 2 ⊢ (¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1〈0R, 1R〉𝑦) ∧ 𝐹∅(0 · 1))) |
36 | 19, 35 | mpbir 233 | 1 ⊢ ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {csn 4569 〈cop 4575 class class class wbr 5068 × cxp 5555 ℩cio 6314 ‘cfv 6357 (class class class)co 7158 Rcnr 10289 0Rc0r 10290 1Rc1r 10291 ℝcr 10538 0cc0 10539 1c1 10540 ici 10541 · cmul 10544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-mulcl 10601 ax-mulcom 10603 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1rid 10609 ax-cnre 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-i 10548 df-r 10549 |
This theorem is referenced by: (None) |
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