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Mirrors > Home > MPE Home > Th. List > 0domg | Structured version Visualization version GIF version |
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5354, ax-un 7719. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
0domg | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5298 | . . 3 ⊢ ∅ ∈ V | |
2 | f1eq1 6773 | . . 3 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
3 | f10 6857 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
4 | 1, 2, 3 | ceqsexv2d 3521 | . 2 ⊢ ∃𝑓 𝑓:∅–1-1→𝐴 |
5 | brdom2g 8948 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1→𝐴)) | |
6 | 1, 5 | mpan 687 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1→𝐴)) |
7 | 4, 6 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wex 1773 ∈ wcel 2098 Vcvv 3466 ∅c0 4315 class class class wbr 5139 –1-1→wf1 6531 ≼ cdom 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-dom 8938 |
This theorem is referenced by: dom0OLD 9100 0sdomg 9101 0sdomgOLD 9102 0dom 9103 sdom0OLD 9106 carddomi2 9962 wdomfil 10053 wdomnumr 10056 hashge0 14348 ufildom1 23774 harn0 42394 safesnsupfidom1o 42717 sn1dom 42826 |
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