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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 5365, ax-un 7755. (Revised by BTernaryTau, 29-Nov-2024.) |
| Ref | Expression |
|---|---|
| 0sdomg | ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0domg 9140 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | |
| 2 | brsdom 9015 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
| 3 | 2 | baib 535 | . . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
| 4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
| 5 | en0r 9060 | . . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) | |
| 6 | 5 | necon3bbii 2988 | . 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 2108 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 df-dom 8987 df-sdom 8988 |
| This theorem is referenced by: 0sdom 9147 fodomr 9168 pwdom 9169 0sdom1dom 9274 sdom1OLD 9279 1sdom2dom 9283 infn0ALT 9341 fodomfir 9368 fodomfib 9369 fodomfibOLD 9371 domwdom 9614 iunfictbso 10154 djulepw 10233 fin45 10432 fodomb 10566 brdom3 10568 gchxpidm 10709 inar1 10815 csdfil 23902 ovoliunnul 25542 carsgclctunlem3 34322 domalom 37405 ovoliunnfl 37669 voliunnfl 37671 volsupnfl 37672 sdomne0 43426 sdomne0d 43427 ensucne0OLD 43543 nnfoctb 45053 caragenunicl 46539 |
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