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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 9159 | . . . . 5 ⊢ ((𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐵 ≈ 𝐴) | |
2 | 1 | expcom 413 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴)) |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 → (𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴))) |
4 | iman 401 | . . . 4 ⊢ ((𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴) ↔ ¬ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴)) | |
5 | brsdom 9035 | . . . 4 ⊢ (𝐵 ≺ 𝐴 ↔ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴)) | |
6 | 4, 5 | xchbinxr 335 | . . 3 ⊢ ((𝐵 ≼ 𝐴 → 𝐵 ≈ 𝐴) ↔ ¬ 𝐵 ≺ 𝐴) |
7 | 3, 6 | imbitrdi 251 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)) |
8 | onelss 6437 | . . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) | |
9 | ssdomg 9060 | . . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | |
10 | 8, 9 | syld 47 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ≼ 𝐵)) |
11 | 10 | adantl 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ≼ 𝐵)) |
12 | 11 | con3d 152 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → ¬ 𝐴 ∈ 𝐵)) |
13 | ontri1 6429 | . . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) | |
14 | 13 | ancoms 458 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) |
15 | 12, 14 | sylibrd 259 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ⊆ 𝐴)) |
16 | ssdomg 9060 | . . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
17 | 16 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) |
18 | 15, 17 | syld 47 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ≼ 𝐴)) |
19 | ensym 9063 | . . . . . . 7 ⊢ (𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵) | |
20 | endom 9039 | . . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ (𝐵 ≈ 𝐴 → 𝐴 ≼ 𝐵) |
22 | 21 | con3i 154 | . . . . 5 ⊢ (¬ 𝐴 ≼ 𝐵 → ¬ 𝐵 ≈ 𝐴) |
23 | 18, 22 | jca2 513 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴))) |
24 | 23, 5 | imbitrrdi 252 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ≺ 𝐴)) |
25 | 24 | con1d 145 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
26 | 7, 25 | impbid 212 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3976 class class class wbr 5166 Oncon0 6395 ≈ cen 9000 ≼ cdom 9001 ≺ csdm 9002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 |
This theorem is referenced by: sdomel 9190 cardsdomel 10043 alephord 10144 alephsucdom 10148 alephdom2 10156 |
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