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Theorem ordtri1 6353

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 6349	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
2		ordn2lp 6340	. . . . 5 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		imnan 399	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
4	2, 3	sylibr 234	. . . 4 ⊢ (Ord 𝐴 → (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴))
5		ordirr 6338	. . . . 5 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6		eleq2 2817	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵))
7	6	notbid 318	. . . . 5 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵))
8	5, 7	syl5ibrcom 247	. . . 4 ⊢ (Ord 𝐵 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴))
9	4, 8	jaao 956	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → ¬ 𝐵 ∈ 𝐴))
10		ordtri3or 6352	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
11		df-3or 1087	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
12	10, 11	sylib 218	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
13	12	orcomd 871	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ∨ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
14	13	ord 864	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
15	9, 14	impbid 212	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ¬ 𝐵 ∈ 𝐴))
16	1, 15	bitrd 279	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1540 ∈ wcel 2109 ⊆ wss 3911 Ord word 6319

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-tr 5210 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-ord 6323

This theorem is referenced by: ontri1 6354 ordtri2 6355 ordtri4 6357 ordtr3 6366 ordintdif 6371 ordtri2or 6420 ordsucss 7773 ordsucsssuc 7778 ordsucuniel 7779 limsssuc 7806 ssnlim 7842 smoword 8312 tfrlem15 8337 nnaword 8568 nnawordex 8578 eldifsucnn 8605 nndomog 9154 onomeneq 9155 isfinite2 9221 unfilem1 9230 wofib 9474 cantnflem1 9618 ttrcltr 9645 dmttrcl 9650 alephgeom 10011 alephdom2 10016 cflim2 10192 fin67 10324 winainflem 10622 finminlem 36279 ordeldif 43220 ordeldifsucon 43221 ordeldif1o 43222

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