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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 8640 | . . 3 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≺ ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ◡ ≼ )) |
3 | df-br 5067 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
4 | df-br 5067 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
5 | df-br 5067 | . . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ 〈𝐵, 𝐴〉 ∈ ≼ ) | |
6 | brsdom2.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
7 | brsdom2.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
8 | 6, 7 | opelcnv 5752 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ ◡ ≼ ↔ 〈𝐵, 𝐴〉 ∈ ≼ ) |
9 | 5, 8 | bitr4i 280 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 ↔ 〈𝐴, 𝐵〉 ∈ ◡ ≼ ) |
10 | 9 | notbii 322 | . . . 4 ⊢ (¬ 𝐵 ≼ 𝐴 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ◡ ≼ ) |
11 | 4, 10 | anbi12i 628 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ◡ ≼ )) |
12 | eldif 3946 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ◡ ≼ ) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ◡ ≼ )) | |
13 | 11, 12 | bitr4i 280 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ◡ ≼ )) |
14 | 2, 3, 13 | 3bitr4i 305 | 1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 〈cop 4573 class class class wbr 5066 ◡ccnv 5554 ≼ cdom 8507 ≺ csdm 8508 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 |
This theorem is referenced by: (None) |
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