Theorem son2lpi 5135

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 5133	. 2 ⊢ ¬ 𝐴𝑅𝐴
4	1, 2	sotri 5134	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴)
5	3, 4	mto 668	1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 104   ⊆ wss 3199   class class class wbr 4089   Or wor 4394   × cxp 4725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-po 4395  df-iso 4396  df-xp 4733

This theorem is referenced by:  nqprdisj  7769  ltexprlemdisj  7831  recexprlemdisj  7855  caucvgprlemnkj  7891  caucvgprprlemnkltj  7914  caucvgprprlemnkeqj  7915  caucvgprprlemnjltk  7916