Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > domtriord | Structured version Visualization version GIF version |
Description: Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.) |
Ref | Expression |
---|---|
domtriord | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbth 7965 | . . . . 5 ⊢ ((𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐵 ≈ 𝐴) | |
2 | 1 | expcom 450 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴)) |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 → (𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴))) |
4 | iman 439 | . . . 4 ⊢ ((𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴) ↔ ¬ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴)) | |
5 | brsdom 7864 | . . . 4 ⊢ (𝐵 ≺ 𝐴 ↔ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴)) | |
6 | 4, 5 | xchbinxr 324 | . . 3 ⊢ ((𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴) ↔ ¬ 𝐵 ≺ 𝐴) |
7 | 3, 6 | syl6ib 240 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)) |
8 | onelss 5683 | . . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) | |
9 | ssdomg 7887 | . . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | |
10 | 8, 9 | syld 46 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ≼ 𝐵)) |
11 | 10 | adantl 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ≼ 𝐵)) |
12 | 11 | con3d 147 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → ¬ 𝐴 ∈ 𝐵)) |
13 | ontri1 5674 | . . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) | |
14 | 13 | ancoms 468 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) |
15 | 12, 14 | sylibrd 248 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ⊆ 𝐴)) |
16 | ssdomg 7887 | . . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
17 | 16 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) |
18 | 15, 17 | syld 46 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ≼ 𝐴)) |
19 | ensym 7891 | . . . . . . . 8 ⊢ (𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵) | |
20 | endom 7868 | . . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ (𝐵 ≈ 𝐴 → 𝐴 ≼ 𝐵) |
22 | 21 | con3i 149 | . . . . . 6 ⊢ (¬ 𝐴 ≼ 𝐵 → ¬ 𝐵 ≈ 𝐴) |
23 | 22 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → ¬ 𝐵 ≈ 𝐴)) |
24 | 18, 23 | jcad 554 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴))) |
25 | 24, 5 | syl6ibr 241 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ≺ 𝐴)) |
26 | 25 | con1d 138 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
27 | 7, 26 | impbid 201 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 Oncon0 5640 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: sdomel 7992 cardsdomel 8683 alephord 8781 alephsucdom 8785 alephdom2 8793 |
Copyright terms: Public domain | W3C validator |