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| Mirrors > Home > MPE Home > Th. List > sdom0OLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of sdom0 9148 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| sdom0OLD | ⊢ ¬ 𝐴 ≺ ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relsdom 8992 | . . . 4 ⊢ Rel ≺ | |
| 2 | 1 | brrelex1i 5741 | . . 3 ⊢ (𝐴 ≺ ∅ → 𝐴 ∈ V) |
| 3 | 0domg 9140 | . . 3 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≺ ∅ → ∅ ≼ 𝐴) |
| 5 | domnsym 9139 | . . 3 ⊢ (∅ ≼ 𝐴 → ¬ 𝐴 ≺ ∅) | |
| 6 | 5 | con2i 139 | . 2 ⊢ (𝐴 ≺ ∅ → ¬ ∅ ≼ 𝐴) |
| 7 | 4, 6 | pm2.65i 194 | 1 ⊢ ¬ 𝐴 ≺ ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ≼ 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-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 |
| This theorem is referenced by: (None) |
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