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Theorem ordtri3 6371

Description: A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.)

**Assertion**
Ref	Expression
ordtri3	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))

**Proof of Theorem ordtri3**
Step	Hyp	Ref	Expression
1		ordirr 6353	. . . . . 6 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
2	1	adantl 481	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵 ∈ 𝐵)
3		eleq2 2818	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵))
4	3	notbid 318	. . . . 5 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵))
5	2, 4	syl5ibrcom 247	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴))
6	5	pm4.71d 561	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
7		pm5.61 1002	. . . 4 ⊢ (((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∧ ¬ 𝐵 ∈ 𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
8		pm4.52 986	. . . 4 ⊢ (((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴))
9	7, 8	bitr3i 277	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴))
10	6, 9	bitrdi 287	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)))
11		ordtri2 6370	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
12	11	orbi1d 916	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)))
13	12	notbid 318	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)))
14	10, 13	bitr4d 282	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 Ord word 6334

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 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

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 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-tr 5218 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-ord 6338

This theorem is referenced by: ordunisuc2 7823 tz7.48lem 8412 oacan 8515 omcan 8536 oecan 8556 omsmo 8625 omopthi 8628 inf3lem6 9593 cantnfp1lem3 9640 infpssrlem5 10267 fin23lem24 10282 isf32lem4 10316 om2uzf1oi 13925 om2noseqf1o 28202 ordnexbtwnsuc 43263

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