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Theorem 0wdom 9514
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 2733 . . 3 ∅ = ∅
21orci 864 . 2 (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)
3 brwdom 9511 . 2 (𝑋𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)))
42, 3mpbiri 258 1 (𝑋𝑉 → ∅ ≼* 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1542  wex 1782  wcel 2107  c0 4286   class class class wbr 5109  ontowfo 6498  * cwdom 9508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-fn 6503  df-fo 6506  df-wdom 9509
This theorem is referenced by:  brwdom2  9517  wdomtr  9519
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