Theorem son2lpi 6091

Description: A strict order relation has no 2-cycle loops. (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
son2lpi	⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)

Proof of Theorem son2lpi

Step	Hyp	Ref	Expression
1		soi.1	. . 3 ⊢ 𝑅 Or 𝑆
2		soi.2	. . 3 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
3	1, 2	soirri 6089	. 2 ⊢ ¬ 𝐴𝑅𝐴
4	1, 2	sotri 6090	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴)
5	3, 4	mto 197	1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   ⊆ wss 3889   class class class wbr 5085   Or wor 5538   × cxp 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-po 5539  df-so 5540  df-xp 5637

This theorem is referenced by: (None)