Theorem 0wdom 9531

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 2761	. . 3 ⊢ ∅ = ∅
2	1	orci 878	. 2 ⊢ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)
3		brwdom 9528	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)))
4	2, 3	mpbiri 261	1 ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∅c0 4285   class class class wbr 5108  –onto→wfo 6534   ≼^* cwdom 9525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-fn 6539  df-fo 6542  df-wdom 9526

This theorem is referenced by:  brwdom2  9534  wdomtr  9536