Theorem avril1 8723

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 germanus 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.)

A reply to skeptics can be found at http://us.metamath.org/mpegif/mmnotes.txt, under the 1-Apr-2006 entry.

Assertion

Ref	Expression
avril1	⊢ ¬ (A℘ℝ(i ‘1) ⋀ F∅(0 · 1))

Proof of Theorem avril1

Step	Hyp	Ref	Expression
1		equid 1124	. . . . . . . 8 ⊢ x = x
2		dfnul2 2278	. . . . . . . . . 10 ⊢ ∅ = {x∣ ¬ x = x}
3	2	abeq2i 1567	. . . . . . . . 9 ⊢ (x ∈ ∅ ↔ ¬ x = x)
4	3	con2bii 221	. . . . . . . 8 ⊢ (x = x ↔ ¬ x ∈ ∅)
5	1, 4	mpbi 189	. . . . . . 7 ⊢ ¬ x ∈ ∅
6		eleq1 1531	. . . . . . 7 ⊢ (x = ⟨F, 0⟩ → (x ∈ ∅ ↔ ⟨F, 0⟩ ∈ ∅))
7	5, 6	mtbii 715	. . . . . 6 ⊢ (x = ⟨F, 0⟩ → ¬ ⟨F, 0⟩ ∈ ∅)
8	7	vtocleg 1851	. . . . 5 ⊢ (⟨F, 0⟩ ∈ V → ¬ ⟨F, 0⟩ ∈ ∅)
9		elisset 1813	. . . . . 6 ⊢ (⟨F, 0⟩ ∈ ∅ → ⟨F, 0⟩ ∈ V)
10	9	con3i 98	. . . . 5 ⊢ (¬ ⟨F, 0⟩ ∈ V → ¬ ⟨F, 0⟩ ∈ ∅)
11	8, 10	pm2.61i 126	. . . 4 ⊢ ¬ ⟨F, 0⟩ ∈ ∅
12		df-br 2615	. . . . 5 ⊢ (F∅(0 · 1) ↔ ⟨F, (0 · 1)⟩ ∈ ∅)
13		0cn 5308	. . . . . . . 8 ⊢ 0 ∈ ℂ
14	13	mulid1 5312	. . . . . . 7 ⊢ (0 · 1) = 0
15	14	opeq2i 2487	. . . . . 6 ⊢ ⟨F, (0 · 1)⟩ = ⟨F, 0⟩
16	15	eleq1i 1534	. . . . 5 ⊢ (⟨F, (0 · 1)⟩ ∈ ∅ ↔ ⟨F, 0⟩ ∈ ∅)
17	12, 16	bitr 173	. . . 4 ⊢ (F∅(0 · 1) ↔ ⟨F, 0⟩ ∈ ∅)
18	11, 17	mtbir 192	. . 3 ⊢ ¬ F∅(0 · 1)
19	18	intnan 690	. 2 ⊢ ¬ (A℘(R × {0R})∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}} ⋀ F∅(0 · 1))
20		df-i 5223	. . . . . . . 8 ⊢ i = ⟨0R, 1R⟩
21	20	fveq1i 3716	. . . . . . 7 ⊢ (i ‘1) = (⟨0R, 1R⟩ ‘1)
22		df-fv 3193	. . . . . . 7 ⊢ (⟨0R, 1R⟩ ‘1) = ∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}}
23	21, 22	eqtr 1492	. . . . . 6 ⊢ (i ‘1) = ∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}}
24	23	breq2i 2622	. . . . 5 ⊢ (A℘ℝ(i ‘1) ↔ A℘ℝ∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}})
25		df-r 5224	. . . . . . 7 ⊢ ℝ = (R × {0R})
26		sseq2 2079	. . . . . . . . 9 ⊢ (ℝ = (R × {0R}) → (z ⊆ ℝ ↔ z ⊆ (R × {0R})))
27	26	abbidv 1574	. . . . . . . 8 ⊢ (ℝ = (R × {0R}) → {z∣z ⊆ ℝ} = {z∣z ⊆ (R × {0R})})
28		df-pw 2398	. . . . . . . 8 ⊢ ℘ℝ = {z∣z ⊆ ℝ}
29		df-pw 2398	. . . . . . . 8 ⊢ ℘(R × {0R}) = {z∣z ⊆ (R × {0R})}
30	27, 28, 29	3eqtr4g 1528	. . . . . . 7 ⊢ (ℝ = (R × {0R}) → ℘ℝ = ℘(R × {0R}))
31	25, 30	ax-mp 7	. . . . . 6 ⊢ ℘ℝ = ℘(R × {0R})
32	31	breqi 2620	. . . . 5 ⊢ (A℘ℝ∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}} ↔ A℘(R × {0R})∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}})
33	24, 32	bitr 173	. . . 4 ⊢ (A℘ℝ(i ‘1) ↔ A℘(R × {0R})∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}})
34	33	anbi1i 481	. . 3 ⊢ ((A℘ℝ(i ‘1) ⋀ F∅(0 · 1)) ↔ (A℘(R × {0R})∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}} ⋀ F∅(0 · 1)))
35	34	negbii 187	. 2 ⊢ (¬ (A℘ℝ(i ‘1) ⋀ F∅(0 · 1)) ↔ ¬ (A℘(R × {0R})∪{y∣(⟨0R, 1R⟩ “ {1}) = {y}} ⋀ F∅(0 · 1)))
36	19, 35	mpbir 190	1 ⊢ ¬ (A℘ℝ(i ‘1) ⋀ F∅(0 · 1))

Syntax hints:  ¬ wn 2   ⋀ wa 223   = wceq 954   ∈ wcel 956  {cab 1461  Vcvv 1807   ⊆ wss 2043  ∅c0 2276  ℘cpw 2397  {csn 2405  ⟨cop 2407  ∪cuni 2498   class class class wbr 2614   × cxp 3163   “ cima 3168   ‘cfv 3177  (class class class)co 3954  Rcnr 4973  0_Rc0r 4974  1_Rc1r 4975  ℝcr 5213  0cc0 5214  1c1 5215  ici 5216   · cmul 5219

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226

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