Theorem 0wdom 9476

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 2737	. . 3 ⊢ ∅ = ∅
2	1	orci 866	. 2 ⊢ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)
3		brwdom 9473	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅)))
4	2, 3	mpbiri 258	1 ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋)

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∅c0 4274   class class class wbr 5086  –onto→wfo 6488   ≼^* cwdom 9470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-fn 6493  df-fo 6496  df-wdom 9471

This theorem is referenced by:  brwdom2  9479  wdomtr  9481