Theorem sotri 5442

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 5090	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3	2	simpld 474	. . 3 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ 𝑆)
4	1	brel 5090	. . 3 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
5	3, 4	anim12i 588	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
6		soi.1	. . . 4 ⊢ 𝑅 Or 𝑆
7		sotr 4981	. . . 4 ⊢ ((𝑅 Or 𝑆 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
8	6, 7	mpan 702	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
9	8	3expb 1258	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
10	5, 9	mpcom 37	1 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583   Or wor 4958   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-xp 5044

This theorem is referenced by:  son2lpi  5443  sotri2  5444  sotri3  5445  ltsonq  9670  ltbtwnnq  9679  nqpr  9715  prlem934  9734  ltexprlem4  9740  reclem2pr  9749  reclem4pr  9751  ltsosr  9794  addgt0sr  9804  supsrlem  9811  axpre-lttrn  9866