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| Mirrors > Home > MPE Home > Th. List > brsdom2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.) |
| Ref | Expression |
|---|---|
| brsdom2.1 | ⊢ 𝐴 ∈ V |
| brsdom2.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| brsdom2 | ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfsdom2 9136 | . . 3 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≺ ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ◡ ≼ )) |
| 3 | df-br 5144 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
| 4 | df-br 5144 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
| 5 | df-br 5144 | . . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ 〈𝐵, 𝐴〉 ∈ ≼ ) | |
| 6 | brsdom2.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
| 7 | brsdom2.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
| 8 | 6, 7 | opelcnv 5892 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ ◡ ≼ ↔ 〈𝐵, 𝐴〉 ∈ ≼ ) |
| 9 | 5, 8 | bitr4i 278 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 ↔ 〈𝐴, 𝐵〉 ∈ ◡ ≼ ) |
| 10 | 9 | notbii 320 | . . . 4 ⊢ (¬ 𝐵 ≼ 𝐴 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ◡ ≼ ) |
| 11 | 4, 10 | anbi12i 628 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ◡ ≼ )) |
| 12 | eldif 3961 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ◡ ≼ ) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ◡ ≼ )) | |
| 13 | 11, 12 | bitr4i 278 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ◡ ≼ )) |
| 14 | 2, 3, 13 | 3bitr4i 303 | 1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 〈cop 4632 class class class wbr 5143 ◡ccnv 5684 ≼ 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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