Theorem avril1 4523

Description: Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting." (Contributed by Loof Lirpa 1-Apr-04.)

Assertion

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

Proof of Theorem avril1

Step	Hyp	Ref	Expression
1		noel 1711	. . 3 ⊢ ¬ ⟨F, (0 · 1)⟩ ∈ ∅
2		df-br 2063	. . 3 ⊢ (F∅(0 · 1) ↔ ⟨F, (0 · 1)⟩ ∈ ∅)
3	1, 2	mtbir 167	. 2 ⊢ ¬ F∅(0 · 1)
4	3	intnan 516	1 ⊢ ¬ (A℘R(i ‘1) ∧ F∅(0 · 1))

Syntax hints:  ¬ wn 1   ∧ wa 196   ∈ wcel 1092  ∅c0 1707  ℘cpw 1798  ⟨cop 1810   class class class wbr 2054   ‘cfv 2422  (class class class)co 3001  0cc0 4028  1c1 4029  ici 4030   · cmulc 4032

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-nul 1708  df-br 2063

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