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Theorem 0wdom 8420
 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 2626 . . 3 ∅ = ∅
21orci 405 . 2 (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)
3 brwdom 8417 . 2 (𝑋𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋onto→∅)))
42, 3mpbiri 248 1 (𝑋𝑉 → ∅ ≼* 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   = wceq 1480  ∃wex 1701   ∈ wcel 1992  ∅c0 3896   class class class wbr 4618  –onto→wfo 5848   ≼* cwdom 8407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-fn 5853  df-fo 5856  df-wdom 8409 This theorem is referenced by:  brwdom2  8423  wdomtr  8425
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