Theorem 0wdom 9456

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 2731	. . 3 ⊢ ∅ = ∅
2	1	orci 865	. 2 ⊢ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)
3		brwdom 9453	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)))
4	2, 3	mpbiri 258	1 ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∅c0 4280   class class class wbr 5089  –onto→wfo 6479   ≼^* cwdom 9450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-fn 6484  df-fo 6487  df-wdom 9451

This theorem is referenced by:  brwdom2  9459  wdomtr  9461