Theorem ordtri3 2234

Description: A trichotomy law for ordinals.

Assertion

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

Proof of Theorem ordtri3

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . . . . 7 ⊢ (A = B → (A ∈ A ↔ A ∈ B))
2	1	negbid 463	. . . . . 6 ⊢ (A = B → (¬ A ∈ A ↔ ¬ A ∈ B))
3		ordeirr 2217	. . . . . 6 ⊢ (Ord A → ¬ A ∈ A)
4	2, 3	syl5bi 183	. . . . 5 ⊢ (A = B → (Ord A → ¬ A ∈ B))
5		eleq2 1150	. . . . . . 7 ⊢ (A = B → (B ∈ A ↔ B ∈ B))
6	5	negbid 463	. . . . . 6 ⊢ (A = B → (¬ B ∈ A ↔ ¬ B ∈ B))
7		ordeirr 2217	. . . . . 6 ⊢ (Ord B → ¬ B ∈ B)
8	6, 7	syl5bir 184	. . . . 5 ⊢ (A = B → (Ord B → ¬ B ∈ A))
9	4, 8	anim12d 431	. . . 4 ⊢ (A = B → ((Ord A ∧ Ord B) → (¬ A ∈ B ∧ ¬ B ∈ A)))
10		ioran 254	. . . 4 ⊢ (¬ (A ∈ B ∨ B ∈ A) ↔ (¬ A ∈ B ∧ ¬ B ∈ A))
11	9, 10	syl6ibr 186	. . 3 ⊢ (A = B → ((Ord A ∧ Ord B) → ¬ (A ∈ B ∨ B ∈ A)))
12	11	com12 13	. 2 ⊢ ((Ord A ∧ Ord B) → (A = B → ¬ (A ∈ B ∨ B ∈ A)))
13		ordtri3or 2230	. . 3 ⊢ ((Ord A ∧ Ord B) → (A ∈ B ∨ A = B ∨ B ∈ A))
14		df-3or 582	. . . 4 ⊢ ((A ∈ B ∨ A = B ∨ B ∈ A) ↔ ((A ∈ B ∨ A = B) ∨ B ∈ A))
15		or23 219	. . . 4 ⊢ (((A ∈ B ∨ A = B) ∨ B ∈ A) ↔ ((A ∈ B ∨ B ∈ A) ∨ A = B))
16		df-or 197	. . . 4 ⊢ (((A ∈ B ∨ B ∈ A) ∨ A = B) ↔ (¬ (A ∈ B ∨ B ∈ A) → A = B))
17	14, 15, 16	3bitr 155	. . 3 ⊢ ((A ∈ B ∨ A = B ∨ B ∈ A) ↔ (¬ (A ∈ B ∨ B ∈ A) → A = B))
18	13, 17	sylib 173	. 2 ⊢ ((Ord A ∧ Ord B) → (¬ (A ∈ B ∨ B ∈ A) → A = B))
19	12, 18	impbid 397	1 ⊢ ((Ord A ∧ Ord B) → (A = B ↔ ¬ (A ∈ B ∨ B ∈ A)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∨ w3o 580   = wceq 1091   ∈ wcel 1092  Ord word 2198

This theorem is referenced by:  ordtri4 2235  tz7.48lem 2993  oacan 3150  nnmcan 3190  omsmo 3196  inf3lem6 3469  om2uzf1o 4656

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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