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| 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 9021 | . . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
| 2 | domtr 9048 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
| 3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | 
| 4 | simpl 482 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐵) | |
| 5 | simpr 484 | . . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐵 ≼ 𝐶) | |
| 6 | ensym 9044 | . . . . . 6 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ≈ 𝐴) | |
| 7 | domentr 9054 | . . . . . 6 ⊢ ((𝐵 ≼ 𝐶 ∧ 𝐶 ≈ 𝐴) → 𝐵 ≼ 𝐴) | |
| 8 | 5, 6, 7 | syl2an 596 | . . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) ∧ 𝐴 ≈ 𝐶) → 𝐵 ≼ 𝐴) | 
| 9 | domnsym 9140 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
| 10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) ∧ 𝐴 ≈ 𝐶) → ¬ 𝐴 ≺ 𝐵) | 
| 11 | 10 | ex 412 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → (𝐴 ≈ 𝐶 → ¬ 𝐴 ≺ 𝐵)) | 
| 12 | 4, 11 | mt2d 136 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → ¬ 𝐴 ≈ 𝐶) | 
| 13 | brsdom 9016 | . 2 ⊢ (𝐴 ≺ 𝐶 ↔ (𝐴 ≼ 𝐶 ∧ ¬ 𝐴 ≈ 𝐶)) | |
| 14 | 3, 12, 13 | sylanbrc 583 | 1 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 class class class wbr 5142 ≈ cen 8983 ≼ cdom 8984 ≺ csdm 8985 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 | 
| This theorem is referenced by: sdomentr 9152 sucdomOLD 9273 infsdomnnOLD 9340 fodomfibOLD 9372 marypha1lem 9474 r1sdom 9815 infxpenlem 10054 infunsdom1 10253 fin56 10434 fodomb 10567 pwcfsdom 10624 cfpwsdom 10625 canthp1lem2 10694 gchpwdom 10711 gchhar 10720 gchina 10740 tsksdom 10797 tskpr 10811 tskcard 10822 gruina 10859 domalom 37406 lindsenlbs 37623 | 
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