Theorem son2lpi 6085

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 6083	. 2 ⊢ ¬ 𝐴𝑅𝐴
4	1, 2	sotri 6084	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴)
5	3, 4	mto 198	1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 396   ⊆ wss 3890   class class class wbr 5079   Or wor 5532   × cxp 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-po 5533  df-so 5534  df-xp 5631

This theorem is referenced by: (None)