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Theorem 0wdom 9022
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 2818 . . 3 ∅ = ∅
21orci 859 . 2 (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)
3 brwdom 9019 . 2 (𝑋𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)))
42, 3mpbiri 259 1 (𝑋𝑉 → ∅ ≼* 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 841   = wceq 1528  wex 1771  wcel 2105  c0 4288   class class class wbr 5057  ontowfo 6346  * cwdom 9009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-fn 6351  df-fo 6354  df-wdom 9011
This theorem is referenced by:  brwdom2  9025  wdomtr  9027
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