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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 5371, ax-un 7754. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
dom0 | ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 8998 | . . 3 ⊢ (𝐴 ≼ ∅ → ∃𝑓 𝑓:𝐴–1-1→∅) | |
2 | f1f 6805 | . . . . 5 ⊢ (𝑓:𝐴–1-1→∅ → 𝑓:𝐴⟶∅) | |
3 | f00 6791 | . . . . . 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 1928 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→∅ → 𝐴 = ∅) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝐴 ≼ ∅ → 𝐴 = ∅) |
8 | en0 9057 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
9 | endom 9018 | . . 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 1537 ∃wex 1776 ∅c0 4339 class class class wbr 5148 ⟶wf 6559 –1-1→wf1 6560 ≈ cen 8981 ≼ cdom 8982 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-en 8985 df-dom 8986 |
This theorem is referenced by: sdom0 9147 0sdom1dom 9272 fin1a2lem11 10448 cfpwsdom 10622 |
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