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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 9093 | . . 3 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | |
2 | 1 | eleq2i 2826 | . 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 5880 | . . . . . 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 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 397 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3945 〈cop 4634 class class class wbr 5148 ◡ccnv 5675 ≼ cdom 8934 ≺ csdm 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 |
This theorem is referenced by: (None) |
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