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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.) |
Ref | Expression |
---|---|
dom0 | ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 7961 | . . . . 5 ⊢ Rel ≼ | |
2 | 1 | brrelexi 5158 | . . . 4 ⊢ (𝐴 ≼ ∅ → 𝐴 ∈ V) |
3 | 0domg 8087 | . . . 4 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ≼ ∅ → ∅ ≼ 𝐴) |
5 | 4 | pm4.71i 664 | . 2 ⊢ (𝐴 ≼ ∅ ↔ (𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴)) |
6 | sbthb 8081 | . 2 ⊢ ((𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴) ↔ 𝐴 ≈ ∅) | |
7 | en0 8019 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
8 | 5, 6, 7 | 3bitri 286 | 1 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 class class class wbr 4653 ≈ cen 7952 ≼ cdom 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-er 7742 df-en 7956 df-dom 7957 |
This theorem is referenced by: pwcdadom 9038 fin1a2lem11 9232 cfpwsdom 9406 |
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