Theorem ordtri3 6399

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 6380	. . . . . 6 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
2	1	adantl 487	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵 ∈ 𝐵)
3		eleq2 2850	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵))
4	3	notbid 321	. . . . 5 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵))
5	2, 4	syl5ibrcom 250	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴))
6	5	pm4.71d 571	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
7		pm5.61 1016	. . . 4 ⊢ (((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∧ ¬ 𝐵 ∈ 𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
8		pm4.52 1000	. . . 4 ⊢ (((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴))
9	7, 8	bitr3i 280	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴))
10	6, 9	bitrdi 290	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)))
11		ordtri2 6398	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
12	11	orbi1d 930	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)))
13	12	notbid 321	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)))
14	10, 13	bitr4d 285	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Ord word 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365

This theorem is used by:  ordunisuc2  7855  onelfvnef1  8449  tz7.48lemOLD  8451  oacan  8556  omcan  8577  oecan  8598  omsmo  8667  omopthi  8670  inf3lem6  9634  cantnfp1lem3  9681  infpssrlem5  10385  fin23lem24  10400  isf32lem4  10434  om2uzf1oi  14096  om2noseqf1o  28687  ordnexbtwnsuc  44268