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Mirrors > Home > MPE Home > Th. List > sdomdomtr | Structured version Visualization version GIF version |
Description: Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
sdomdomtr | ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 7869 | . . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domtr 7895 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan 487 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
4 | simpl 472 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐵) | |
5 | simpr 476 | . . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐵 ≼ 𝐶) | |
6 | ensym 7891 | . . . . . 6 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ≈ 𝐴) | |
7 | domentr 7901 | . . . . . 6 ⊢ ((𝐵 ≼ 𝐶 ∧ 𝐶 ≈ 𝐴) → 𝐵 ≼ 𝐴) | |
8 | 5, 6, 7 | syl2an 493 | . . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) ∧ 𝐴 ≈ 𝐶) → 𝐵 ≼ 𝐴) |
9 | domnsym 7971 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) ∧ 𝐴 ≈ 𝐶) → ¬ 𝐴 ≺ 𝐵) |
11 | 10 | ex 449 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → (𝐴 ≈ 𝐶 → ¬ 𝐴 ≺ 𝐵)) |
12 | 4, 11 | mt2d 130 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → ¬ 𝐴 ≈ 𝐶) |
13 | brsdom 7864 | . 2 ⊢ (𝐴 ≺ 𝐶 ↔ (𝐴 ≼ 𝐶 ∧ ¬ 𝐴 ≈ 𝐶)) | |
14 | 3, 12, 13 | sylanbrc 695 | 1 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 class class class wbr 4583 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: sdomentr 7979 sucdom 8042 infsdomnn 8106 fodomfib 8125 marypha1lem 8222 r1sdom 8520 infxpenlem 8719 infunsdom1 8918 fin56 9098 fodomb 9229 pwcfsdom 9284 cfpwsdom 9285 canthp1lem2 9354 gchpwdom 9371 gchhar 9380 gchina 9400 tsksdom 9457 tskpr 9471 tskcard 9482 gruina 9519 lindsenlbs 32574 |
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