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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 5365, ax-un 7755. (Revised by BTernaryTau, 29-Nov-2024.) |
| Ref | Expression |
|---|---|
| 0domg | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ex 5307 | . . 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 3533 | . 2 ⊢ ∃𝑓 𝑓:∅–1-1→𝐴 |
| 5 | brdom2g 8996 | . . 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 1779 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 class class class wbr 5143 –1-1→wf1 6558 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-dom 8987 |
| This theorem is referenced by: dom0OLD 9143 0sdomg 9144 0sdomgOLD 9145 0dom 9146 sdom0OLD 9149 carddomi2 10010 wdomfil 10101 wdomnumr 10104 hashge0 14426 ufildom1 23934 harn0 43114 safesnsupfidom1o 43430 sn1dom 43539 |
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