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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 8917 | . . 3 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | |
2 | 1 | eleq2i 2828 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ◡ ≼ )) |
3 | df-br 5082 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ ) | |
4 | df-br 5082 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ ) | |
5 | df-br 5082 | . . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ ) | |
6 | brsdom2.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
7 | brsdom2.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
8 | 6, 7 | opelcnv 5799 | . . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ◡ ≼ ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ ) |
9 | 5, 8 | bitr4i 279 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ◡ ≼ ) |
10 | 9 | notbii 321 | . . . 4 ⊢ (¬ 𝐵 ≼ 𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ◡ ≼ ) |
11 | 4, 10 | anbi12i 628 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ◡ ≼ )) |
12 | eldif 3902 | . . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ◡ ≼ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ◡ ≼ )) | |
13 | 11, 12 | bitr4i 279 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ◡ ≼ )) |
14 | 2, 3, 13 | 3bitr4i 304 | 1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∈ wcel 2104 Vcvv 3437 ∖ cdif 3889 ⟨cop 4571 class class class wbr 5081 ◡ccnv 5595 ≼ cdom 8758 ≺ csdm 8759 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-id 5496 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 |
This theorem is referenced by: (None) |
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