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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 7847 | . . . . 5 ⊢ Rel ≼ | |
2 | 1 | brrelexi 5082 | . . . 4 ⊢ (𝐴 ≼ ∅ → 𝐴 ∈ V) |
3 | 0domg 7972 | . . . 4 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ≼ ∅ → ∅ ≼ 𝐴) |
5 | 4 | pm4.71i 662 | . 2 ⊢ (𝐴 ≼ ∅ ↔ (𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴)) |
6 | sbthb 7966 | . 2 ⊢ ((𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴) ↔ 𝐴 ≈ ∅) | |
7 | en0 7905 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
8 | 5, 6, 7 | 3bitri 285 | 1 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 class class class wbr 4583 ≈ cen 7838 ≼ cdom 7839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-dom 7843 |
This theorem is referenced by: pwcdadom 8921 fin1a2lem11 9115 cfpwsdom 9285 |
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