Theorem noelOLD 4297

Description: Obsolete version of noel 4296 as of 3-May-2023. The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
noelOLD	⊢ ¬ 𝐴 ∈ ∅

Proof of Theorem noelOLD

Step	Hyp	Ref	Expression
1		eldifi 4103	. . 3 ⊢ (𝐴 ∈ (V ∖ V) → 𝐴 ∈ V)
2		eldifn 4104	. . 3 ⊢ (𝐴 ∈ (V ∖ V) → ¬ 𝐴 ∈ V)
3	1, 2	pm2.65i 196	. 2 ⊢ ¬ 𝐴 ∈ (V ∖ V)
4		df-nul 4292	. . 3 ⊢ ∅ = (V ∖ V)
5	4	eleq2i 2904	. 2 ⊢ (𝐴 ∈ ∅ ↔ 𝐴 ∈ (V ∖ V))
6	3, 5	mtbir 325	1 ⊢ ¬ 𝐴 ∈ ∅

Syntax hints:  ¬ wn 3   ∈ wcel 2114  Vcvv 3494   ∖ cdif 3933  ∅c0 4291

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 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-nul 4292

This theorem is referenced by: (None)