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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 9099 | . . 3 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | |
2 | 1 | eleq2i 2824 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ◡ ≼ )) |
3 | df-br 5149 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ ) | |
4 | df-br 5149 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ ) | |
5 | df-br 5149 | . . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ ) | |
6 | brsdom2.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
7 | brsdom2.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
8 | 6, 7 | opelcnv 5881 | . . . . . 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 626 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ◡ ≼ )) |
12 | eldif 3958 | . . 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 205 ∧ wa 395 ∈ wcel 2105 Vcvv 3473 ∖ cdif 3945 ⟨cop 4634 class class class wbr 5148 ◡ccnv 5675 ≼ cdom 8940 ≺ csdm 8941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 |
This theorem is referenced by: (None) |
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