Theorem itlso 2151

Description: A irreflexive, transitive, linear relation is a strict ordering.

Hypotheses

Ref	Expression
itlso.1	⊢ (x ∈ A → ¬ xRx)
itlso.2	⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → ((xRy ∧ yRz) → xRz))
itlso.3	⊢ ((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx))

Assertion

Ref	Expression
itlso	⊢ R Or A

Distinct variable group(s):   x,y,z,R   x,A,y,z

Proof of Theorem itlso

Step	Hyp	Ref	Expression
1		itlso.1	. . . . . . . 8 ⊢ (x ∈ A → ¬ xRx)
2	1	adantr 306	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ A) → ¬ xRx)
3	2	3adant3 599	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → ¬ xRx)
4		itlso.2	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → ((xRy ∧ yRz) → xRz))
5	3, 4	jca 236	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (¬ xRx ∧ ((xRy ∧ yRz) → xRz)))
6	5	rgen3 1265	. . . 4 ⊢ ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ xRx ∧ ((xRy ∧ yRz) → xRz))
7		df-po 2128	. . . 4 ⊢ (R Po A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ xRx ∧ ((xRy ∧ yRz) → xRz)))
8	6, 7	mpbir 165	. . 3 ⊢ R Po A
9		itlso.3	. . . 4 ⊢ ((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx))
10	9	rgen2 1248	. . 3 ⊢ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)
11	8, 10	pm3.2i 234	. 2 ⊢ (R Po A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx))
12		df-so 2138	. 2 ⊢ (R Or A ↔ (R Po A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)))
13	11, 12	mpbir 165	1 ⊢ R Or A

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∨ w3o 580   ∧ w3a 581   = weq 797   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054   Po wpo 2058   Or wor 2059

This theorem is referenced by:  so 2152  ltsopr 3930

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-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-cleq 1097  df-clel 1099  df-ral 1205  df-po 2128  df-so 2138

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