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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 2821 | . . 3 ⊢ ∅ = ∅ | |
2 | 1 | orci 861 | . 2 ⊢ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅) |
3 | brwdom 9030 | . 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ≼* 𝑋 ↔ (∅ = ∅ ∨ ∃𝑧 𝑧:𝑋–onto→∅))) | |
4 | 2, 3 | mpbiri 260 | 1 ⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∅c0 4290 class class class wbr 5065 –onto→wfo 6352 ≼* cwdom 9020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 df-fn 6357 df-fo 6360 df-wdom 9022 |
This theorem is referenced by: brwdom2 9036 wdomtr 9038 |
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