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Mirrors > Home > MPE Home > Th. List > domsdomtr | Structured version Visualization version GIF version |
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 8927 | . . 3 ⊢ (𝐵 ≺ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | domtr 8954 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan2 594 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≼ 𝐶) |
4 | simpr 486 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐵 ≺ 𝐶) | |
5 | ensym 8950 | . . . . . 6 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ≈ 𝐴) | |
6 | simpl 484 | . . . . . 6 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≼ 𝐵) | |
7 | endomtr 8959 | . . . . . 6 ⊢ ((𝐶 ≈ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐶 ≼ 𝐵) | |
8 | 5, 6, 7 | syl2anr 598 | . . . . 5 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) ∧ 𝐴 ≈ 𝐶) → 𝐶 ≼ 𝐵) |
9 | domnsym 9050 | . . . . 5 ⊢ (𝐶 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐶) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) ∧ 𝐴 ≈ 𝐶) → ¬ 𝐵 ≺ 𝐶) |
11 | 10 | ex 414 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → (𝐴 ≈ 𝐶 → ¬ 𝐵 ≺ 𝐶)) |
12 | 4, 11 | mt2d 136 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → ¬ 𝐴 ≈ 𝐶) |
13 | brsdom 8922 | . 2 ⊢ (𝐴 ≺ 𝐶 ↔ (𝐴 ≼ 𝐶 ∧ ¬ 𝐴 ≈ 𝐶)) | |
14 | 3, 12, 13 | sylanbrc 584 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 class class class wbr 5110 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: ensdomtr 9064 sdomtr 9066 2pwuninel 9083 card2on 9497 tskwe 9893 harval2 9940 prdom2 9949 infxpenlem 9956 alephsucdom 10022 pwsdompw 10147 infunsdom1 10156 fin34 10333 ondomon 10506 cardmin 10507 konigthlem 10511 gchpwdom 10613 gchina 10642 inar1 10718 tskord 10723 tskuni 10726 tskurn 10732 csdfil 23261 ctbssinf 35906 pibt2 35917 |
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