Theorem sotri 6118

Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	⊢ 𝑅 Or 𝑆
soi.2	⊢ 𝑅 ⊆ (𝑆 × 𝑆)

Assertion

Ref	Expression
sotri	⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri

Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5731	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3	2	simpld 494	. . 3 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ 𝑆)
4	1	brel 5731	. . 3 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
5	3, 4	anim12i 612	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
6		soi.1	. . . 4 ⊢ 𝑅 Or 𝑆
7		sotr 5602	. . . 4 ⊢ ((𝑅 Or 𝑆 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
8	6, 7	mpan 687	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
9	8	3expb 1117	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
10	5, 9	mpcom 38	1 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   ∈ wcel 2098   ⊆ wss 3940   class class class wbr 5138   Or wor 5577   × cxp 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-po 5578  df-so 5579  df-xp 5672

This theorem is referenced by:  son2lpi  6119  sotri2  6120  sotri3  6121  ltsonq  10960  ltbtwnnq  10969  nqpr  11005  prlem934  11024  ltexprlem4  11030  reclem2pr  11039  reclem4pr  11041  ltsosr  11085  addgt0sr  11095  supsrlem  11102  axpre-lttrn  11157