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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 9135 | . . 3 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | |
2 | 1 | eleq2i 2831 | . 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 5895 | . . . . . 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 3973 | . . 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 2106 Vcvv 3478 ∖ cdif 3960 〈cop 4637 class class class wbr 5148 ◡ccnv 5688 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: (None) |
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