Theorem 0wdom 9591

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 2763	. . 3 ⊢ ∅ = ∅
2	1	orci 879	. 2 ⊢ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋⟶onto∅)
3		brwdom 9588	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋⟶onto∅)))
4	2, 3	mpbiri 261	1 ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∅c0 4281   class class class wbr 5106  ⟶ontowfo 6556   ≼^* cwdom 9585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-fn 6561  df-fo 6564  df-wdom 9586

This theorem is used by:  brwdom2  9594  wdomtr  9596