Theorem 0domgOLD 9141

Description: Obsolete version of 0domg 9140 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
0domgOLD	⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)

Proof of Theorem 0domgOLD

Step	Hyp	Ref	Expression
1		0ss 4400	. 2 ⊢ ∅ ⊆ 𝐴
2		ssdomg 9040	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ⊆ 𝐴 → ∅ ≼ 𝐴))
3	1, 2	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   ≼ cdom 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-dom 8987

This theorem is referenced by: (None)