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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 8525 | . . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domtr 8550 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
4 | simpl 483 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐵) | |
5 | simpr 485 | . . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐵 ≼ 𝐶) | |
6 | ensym 8546 | . . . . . 6 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ≈ 𝐴) | |
7 | domentr 8556 | . . . . . 6 ⊢ ((𝐵 ≼ 𝐶 ∧ 𝐶 ≈ 𝐴) → 𝐵 ≼ 𝐴) | |
8 | 5, 6, 7 | syl2an 595 | . . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) ∧ 𝐴 ≈ 𝐶) → 𝐵 ≼ 𝐴) |
9 | domnsym 8631 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) ∧ 𝐴 ≈ 𝐶) → ¬ 𝐴 ≺ 𝐵) |
11 | 10 | ex 413 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → (𝐴 ≈ 𝐶 → ¬ 𝐴 ≺ 𝐵)) |
12 | 4, 11 | mt2d 138 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → ¬ 𝐴 ≈ 𝐶) |
13 | brsdom 8520 | . 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 396 class class class wbr 5057 ≈ cen 8494 ≼ cdom 8495 ≺ csdm 8496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 |
This theorem is referenced by: sdomentr 8639 sucdom 8703 infsdomnn 8767 fodomfib 8786 marypha1lem 8885 r1sdom 9191 infxpenlem 9427 infunsdom1 9623 fin56 9803 fodomb 9936 pwcfsdom 9993 cfpwsdom 9994 canthp1lem2 10063 gchpwdom 10080 gchhar 10089 gchina 10109 tsksdom 10166 tskpr 10180 tskcard 10191 gruina 10228 domalom 34567 lindsenlbs 34768 |
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