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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 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
0domg | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5269 | . . 3 ⊢ ∅ ∈ V | |
2 | f1eq1 6738 | . . 3 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
3 | f10 6822 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
4 | 1, 2, 3 | ceqsexv2d 3500 | . 2 ⊢ ∃𝑓 𝑓:∅–1-1→𝐴 |
5 | brdom2g 8902 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1→𝐴)) | |
6 | 1, 5 | mpan 689 | . 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 1782 ∈ wcel 2107 Vcvv 3448 ∅c0 4287 class class class wbr 5110 –1-1→wf1 6498 ≼ cdom 8888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-dom 8892 |
This theorem is referenced by: dom0OLD 9054 0sdomg 9055 0sdomgOLD 9056 0dom 9057 sdom0OLD 9060 carddomi2 9913 wdomfil 10004 wdomnumr 10007 hashge0 14294 ufildom1 23293 harn0 41458 safesnsupfidom1o 41763 sn1dom 41872 |
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