![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8972 | . . 3 ⊢ (𝐵 ≺ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | domtr 8999 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan2 593 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≼ 𝐶) |
4 | simpr 485 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐵 ≺ 𝐶) | |
5 | ensym 8995 | . . . . . 6 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ≈ 𝐴) | |
6 | simpl 483 | . . . . . 6 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≼ 𝐵) | |
7 | endomtr 9004 | . . . . . 6 ⊢ ((𝐶 ≈ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐶 ≼ 𝐵) | |
8 | 5, 6, 7 | syl2anr 597 | . . . . 5 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) ∧ 𝐴 ≈ 𝐶) → 𝐶 ≼ 𝐵) |
9 | domnsym 9095 | . . . . 5 ⊢ (𝐶 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐶) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) ∧ 𝐴 ≈ 𝐶) → ¬ 𝐵 ≺ 𝐶) |
11 | 10 | ex 413 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → (𝐴 ≈ 𝐶 → ¬ 𝐵 ≺ 𝐶)) |
12 | 4, 11 | mt2d 136 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → ¬ 𝐴 ≈ 𝐶) |
13 | brsdom 8967 | . 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 5147 ≈ cen 8932 ≼ cdom 8933 ≺ csdm 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 |
This theorem is referenced by: ensdomtr 9109 sdomtr 9111 2pwuninel 9128 card2on 9545 tskwe 9941 harval2 9988 prdom2 9997 infxpenlem 10004 alephsucdom 10070 pwsdompw 10195 infunsdom1 10204 fin34 10381 ondomon 10554 cardmin 10555 konigthlem 10559 gchpwdom 10661 gchina 10690 inar1 10766 tskord 10771 tskuni 10774 tskurn 10780 csdfil 23389 ctbssinf 36275 pibt2 36286 |
Copyright terms: Public domain | W3C validator |