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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 5370, ax-un 7753. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
0domg | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5312 | . . 3 ⊢ ∅ ∈ V | |
2 | f1eq1 6799 | . . 3 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
3 | f10 6881 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
4 | 1, 2, 3 | ceqsexv2d 3532 | . 2 ⊢ ∃𝑓 𝑓:∅–1-1→𝐴 |
5 | brdom2g 8994 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1→𝐴)) | |
6 | 1, 5 | mpan 690 | . 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 206 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 class class class wbr 5147 –1-1→wf1 6559 ≼ cdom 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-dom 8985 |
This theorem is referenced by: dom0OLD 9141 0sdomg 9142 0sdomgOLD 9143 0dom 9144 sdom0OLD 9147 carddomi2 10007 wdomfil 10098 wdomnumr 10101 hashge0 14422 ufildom1 23949 harn0 43090 safesnsupfidom1o 43406 sn1dom 43515 |
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