Theorem po2nr 4345

Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)

Assertion

Ref	Expression
po2nr	⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Proof of Theorem po2nr

Step	Hyp	Ref	Expression
1		poirr 4343	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
2	1	adantrr 479	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵)
3		potr 4344	. . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))
4	3	3exp2 1227	. . . . 5 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
5	4	com34 83	. . . 4 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
6	5	pm2.43d 50	. . 3 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))))
7	6	imp32 257	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))
8	2, 7	mtod 664	1 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2167   class class class wbr 4034   Po wpo 4330

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-po 4332

This theorem is referenced by:  po3nr  4346  so2nr  4357  tridc  6969