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Theorem ordtri1 6396

Description: A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

**Assertion**
Ref	Expression
ordtri1	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))

**Proof of Theorem ordtri1**
Step	Hyp	Ref	Expression
1		ordsseleq 6392	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
2		ordn2lp 6383	. . . . 5 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		imnan 398	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
4	2, 3	sylibr 233	. . . 4 ⊢ (Ord 𝐴 → (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴))
5		ordirr 6381	. . . . 5 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6		eleq2 2820	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵))
7	6	notbid 317	. . . . 5 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵))
8	5, 7	syl5ibrcom 246	. . . 4 ⊢ (Ord 𝐵 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴))
9	4, 8	jaao 951	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → ¬ 𝐵 ∈ 𝐴))
10		ordtri3or 6395	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
11		df-3or 1086	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
12	10, 11	sylib 217	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
13	12	orcomd 867	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ∨ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
14	13	ord 860	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
15	9, 14	impbid 211	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ¬ 𝐵 ∈ 𝐴))
16	1, 15	bitrd 278	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 ∨ w3o 1084 = wceq 1539 ∈ wcel 2104 ⊆ wss 3947 Ord word 6362

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 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-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6366

This theorem is referenced by: ontri1 6397 ordtri2 6398 ordtri4 6400 ordtr3 6408 ordintdif 6413 ordtri2or 6461 ordsucss 7808 ordsucsssuc 7813 ordsucuniel 7814 limsssuc 7841 ssnlim 7877 smoword 8368 tfrlem15 8394 nnaword 8629 nnawordex 8639 eldifsucnn 8665 nndomog 9218 nndomogOLD 9228 onomeneq 9230 onomeneqOLD 9231 isfinite2 9303 unfilem1 9312 wofib 9542 cantnflem1 9686 ttrcltr 9713 dmttrcl 9718 alephgeom 10079 alephdom2 10084 cflim2 10260 fin67 10392 winainflem 10690 finminlem 35506 ordeldif 42310 ordeldifsucon 42311 ordeldif1o 42312

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