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Theorem ordtri3 6361

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 6343	. . . . . 6 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
2	1	adantl 481	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵 ∈ 𝐵)
3		eleq2 2826	. . . . . 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 1003	. . . 4 ⊢ (((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∧ ¬ 𝐵 ∈ 𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
8		pm4.52 987	. . . 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 6360	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
12	11	orbi1d 917	. . 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 848 = wceq 1542 ∈ wcel 2114 Ord word 6324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5243 ax-pr 5379

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-tr 5208 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-ord 6328

This theorem is referenced by: ordunisuc2 7796 tz7.48lem 8382 oacan 8485 omcan 8506 oecan 8527 omsmo 8596 omopthi 8599 inf3lem6 9554 cantnfp1lem3 9601 infpssrlem5 10229 fin23lem24 10244 isf32lem4 10278 om2uzf1oi 13888 om2noseqf1o 28309 ordnexbtwnsuc 43624

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