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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 5359, ax-un 7734. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
0domg | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5301 | . . 3 ⊢ ∅ ∈ V | |
2 | f1eq1 6782 | . . 3 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
3 | f10 6866 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
4 | 1, 2, 3 | ceqsexv2d 3525 | . 2 ⊢ ∃𝑓 𝑓:∅–1-1→𝐴 |
5 | brdom2g 8969 | . . 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 1774 ∈ wcel 2099 Vcvv 3470 ∅c0 4318 class class class wbr 5142 –1-1→wf1 6539 ≼ cdom 8955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-dom 8959 |
This theorem is referenced by: dom0OLD 9121 0sdomg 9122 0sdomgOLD 9123 0dom 9124 sdom0OLD 9127 carddomi2 9987 wdomfil 10078 wdomnumr 10081 hashge0 14372 ufildom1 23823 harn0 42520 safesnsupfidom1o 42841 sn1dom 42950 |
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