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| Mirrors > Home > MPE Home > Th. List > dom0 | Structured version Visualization version GIF version | ||
| Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5365, ax-un 7755. (Revised by BTernaryTau, 29-Nov-2024.) |
| Ref | Expression |
|---|---|
| dom0 | ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brdomi 8999 | . . 3 ⊢ (𝐴 ≼ ∅ → ∃𝑓 𝑓:𝐴–1-1→∅) | |
| 2 | f1f 6804 | . . . . 5 ⊢ (𝑓:𝐴–1-1→∅ → 𝑓:𝐴⟶∅) | |
| 3 | f00 6790 | . . . . . 6 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
| 4 | 3 | simprbi 496 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅) |
| 5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝑓:𝐴–1-1→∅ → 𝐴 = ∅) |
| 6 | 5 | exlimiv 1930 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→∅ → 𝐴 = ∅) |
| 7 | 1, 6 | syl 17 | . 2 ⊢ (𝐴 ≼ ∅ → 𝐴 = ∅) |
| 8 | en0 9058 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
| 9 | endom 9019 | . . 3 ⊢ (𝐴 ≈ ∅ → 𝐴 ≼ ∅) | |
| 10 | 8, 9 | sylbir 235 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≼ ∅) |
| 11 | 7, 10 | impbii 209 | 1 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 ∅c0 4333 class class class wbr 5143 ⟶wf 6557 –1-1→wf1 6558 ≈ cen 8982 ≼ 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-eu 2569 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-fo 6567 df-f1o 6568 df-en 8986 df-dom 8987 |
| This theorem is referenced by: sdom0 9148 0sdom1dom 9274 fin1a2lem11 10450 cfpwsdom 10624 |
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