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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 5371, ax-un 7754. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
0sdomg | ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0domg 9139 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | |
2 | brsdom 9014 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
3 | 2 | baib 535 | . . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
5 | en0r 9059 | . . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) | |
6 | 5 | necon3bbii 2986 | . 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 206 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 class class class wbr 5148 ≈ cen 8981 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: 0sdom 9146 fodomr 9167 pwdom 9168 0sdom1dom 9272 sdom1OLD 9277 1sdom2dom 9281 infn0ALT 9339 fodomfir 9366 fodomfib 9367 fodomfibOLD 9369 domwdom 9612 iunfictbso 10152 djulepw 10231 fin45 10430 fodomb 10564 brdom3 10566 gchxpidm 10707 inar1 10813 csdfil 23918 ovoliunnul 25556 carsgclctunlem3 34302 domalom 37387 ovoliunnfl 37649 voliunnfl 37651 volsupnfl 37652 sdomne0 43403 sdomne0d 43404 ensucne0OLD 43520 nnfoctb 44987 caragenunicl 46480 |
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