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Theorem 0wdom 9031
Description: Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
0wdom (𝑋𝑉 → ∅ ≼* 𝑋)

Proof of Theorem 0wdom
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . 3 ∅ = ∅
21orci 862 . 2 (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)
3 brwdom 9028 . 2 (𝑋𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)))
42, 3mpbiri 261 1 (𝑋𝑉 → ∅ ≼* 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1538  wex 1781  wcel 2115  c0 4276   class class class wbr 5052  ontowfo 6341  * cwdom 9025
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-fn 6346  df-fo 6349  df-wdom 9026
This theorem is referenced by:  brwdom2  9034  wdomtr  9036
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