Theorem ral0OLD 4447

Description: Obsolete version of ral0 4443 as of 2-Sep-2024. (Contributed by NM, 20-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ral0OLD	⊢ ∀𝑥 ∈ ∅ 𝜑

Proof of Theorem ral0OLD

Step	Hyp	Ref	Expression
1		noel 4264	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → 𝜑)
3	2	rgen 3074	1 ⊢ ∀𝑥 ∈ ∅ 𝜑

Syntax hints:   ∈ wcel 2106  ∀wral 3064  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-dif 3890  df-nul 4257

This theorem is referenced by: (None)