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Mirrors > Home > MPE Home > Th. List > 0sdomg | Structured version Visualization version GIF version |
Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
0sdomg | ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0domg 9051 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | |
2 | brsdom 8922 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
3 | 2 | baib 537 | . . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
5 | en0r 8967 | . . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) | |
6 | 5 | necon3bbii 2992 | . 2 ⊢ (¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅) |
7 | 4, 6 | bitrdi 287 | 1 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2107 ≠ wne 2944 ∅c0 4287 class class class wbr 5110 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 |
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-ne 2945 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-fo 6507 df-f1o 6508 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: 0sdom 9058 fodomr 9079 pwdom 9080 0sdom1dom 9189 sdom1OLD 9194 1sdom2dom 9198 infn0ALT 9259 fodomfib 9277 domwdom 9517 iunfictbso 10057 djulepw 10135 fin45 10335 fodomb 10469 brdom3 10471 gchxpidm 10612 inar1 10718 csdfil 23261 ovoliunnul 24887 carsgclctunlem3 32960 domalom 35904 ovoliunnfl 36149 voliunnfl 36151 volsupnfl 36152 sdomne0 41759 sdomne0d 41760 ensucne0OLD 41876 nnfoctb 43329 caragenunicl 44839 |
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