Theorem son2lpi 5000

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 4998	. 2 ⊢ ¬ 𝐴𝑅𝐴
4	1, 2	sotri 4999	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴)
5	3, 4	mto 652	1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 103   ⊆ wss 3116   class class class wbr 3982   Or wor 4273   × cxp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-po 4274  df-iso 4275  df-xp 4610

This theorem is referenced by:  nqprdisj  7485  ltexprlemdisj  7547  recexprlemdisj  7571  caucvgprlemnkj  7607  caucvgprprlemnkltj  7630  caucvgprprlemnkeqj  7631  caucvgprprlemnjltk  7632