Theorem 0domgOLD 9126

Description: Obsolete version of 0domg 9125 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 4398	. 2 ⊢ ∅ ⊆ 𝐴
2		ssdomg 9021	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ⊆ 𝐴 → ∅ ≼ 𝐴))
3	1, 2	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2098   ⊆ wss 3944  ∅c0 4322   class class class wbr 5149   ≼ cdom 8962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-dom 8966

This theorem is referenced by: (None)