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Mirrors > Home > MPE Home > Th. List > sdomtr | Structured version Visualization version GIF version |
Description: Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.) |
Ref | Expression |
---|---|
sdomtr | ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 8539 | . 2 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domsdomtr 8654 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | |
3 | 1, 2 | sylan 582 | 1 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 class class class wbr 5068 ≼ cdom 8509 ≺ csdm 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 |
This theorem is referenced by: sdomn2lp 8658 2pwuninel 8674 2pwne 8675 r1sdom 9205 alephordi 9502 pwsdompw 9628 gruina 10242 rexpen 15583 |
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