Theorem son2lpi 6130

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 6128	. 2 ⊢ ¬ 𝐴𝑅𝐴
4	1, 2	sotri 6129	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴)
5	3, 4	mto 200	1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ⊆ wss 3906   class class class wbr 5111   Or wor 5570   × cxp 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-po 5571  df-so 5572  df-xp 5669

This theorem is used by: (None)