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Mirrors > Home > MPE Home > Th. List > 0wdom | Structured version Visualization version GIF version |
Description: Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
0wdom | ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ ∅ = ∅ | |
2 | 1 | orci 862 | . 2 ⊢ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅) |
3 | brwdom 9015 | . 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅))) | |
4 | 2, 3 | mpbiri 261 | 1 ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∅c0 4243 class class class wbr 5030 –onto→wfo 6322 ≼* cwdom 9012 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-fn 6327 df-fo 6330 df-wdom 9013 |
This theorem is referenced by: brwdom2 9021 wdomtr 9023 |
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