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| Description: Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| domsdomtr | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sdomdom 9021 | . . 3 ⊢ (𝐵 ≺ 𝐶 → 𝐵 ≼ 𝐶) | |
| 2 | domtr 9048 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
| 3 | 1, 2 | sylan2 593 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≼ 𝐶) | 
| 4 | simpr 484 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐵 ≺ 𝐶) | |
| 5 | ensym 9044 | . . . . . 6 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ≈ 𝐴) | |
| 6 | simpl 482 | . . . . . 6 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≼ 𝐵) | |
| 7 | endomtr 9053 | . . . . . 6 ⊢ ((𝐶 ≈ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐶 ≼ 𝐵) | |
| 8 | 5, 6, 7 | syl2anr 597 | . . . . 5 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) ∧ 𝐴 ≈ 𝐶) → 𝐶 ≼ 𝐵) | 
| 9 | domnsym 9140 | . . . . 5 ⊢ (𝐶 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐶) | |
| 10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) ∧ 𝐴 ≈ 𝐶) → ¬ 𝐵 ≺ 𝐶) | 
| 11 | 10 | ex 412 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → (𝐴 ≈ 𝐶 → ¬ 𝐵 ≺ 𝐶)) | 
| 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: ensdomtr 9154 sdomtr 9156 2pwuninel 9173 card2on 9595 tskwe 9991 harval2 10038 prdom2 10047 infxpenlem 10054 alephsucdom 10120 pwsdompw 10244 infunsdom1 10253 fin34 10431 ondomon 10604 cardmin 10605 konigthlem 10609 gchpwdom 10711 gchina 10740 inar1 10816 tskord 10821 tskuni 10824 tskurn 10830 csdfil 23903 ctbssinf 37408 pibt2 37419 | 
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