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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 5364, ax-un 7725. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
0sdomg | ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0domg 9100 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | |
2 | brsdom 8971 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
3 | 2 | baib 537 | . . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
5 | en0r 9016 | . . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) | |
6 | 5 | necon3bbii 2989 | . 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 2941 ∅c0 4323 class class class wbr 5149 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-en 8940 df-dom 8941 df-sdom 8942 |
This theorem is referenced by: 0sdom 9107 fodomr 9128 pwdom 9129 0sdom1dom 9238 sdom1OLD 9243 1sdom2dom 9247 infn0ALT 9308 fodomfib 9326 domwdom 9569 iunfictbso 10109 djulepw 10187 fin45 10387 fodomb 10521 brdom3 10523 gchxpidm 10664 inar1 10770 csdfil 23398 ovoliunnul 25024 carsgclctunlem3 33319 domalom 36285 ovoliunnfl 36530 voliunnfl 36532 volsupnfl 36533 sdomne0 42164 sdomne0d 42165 ensucne0OLD 42281 nnfoctb 43734 caragenunicl 45240 |
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