Theorem 0wdom 9592

Description: Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Assertion

Ref	Expression
0wdom	⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋)

Proof of Theorem 0wdom

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ ∅ = ∅
2	1	orci 865	. 2 ⊢ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)
3		brwdom 9589	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)))
4	2, 3	mpbiri 258	1 ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∅c0 4313   class class class wbr 5123  –onto→wfo 6539   ≼^* cwdom 9586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-fn 6544  df-fo 6547  df-wdom 9587

This theorem is referenced by:  brwdom2  9595  wdomtr  9597