Theorem sonr 5085

Description: A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)

Assertion

Ref	Expression
sonr	⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem sonr

Step	Hyp	Ref	Expression
1		sopo 5081	. 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
2		poirr 5075	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
3	1, 2	sylan 487	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 2030   class class class wbr 4685   Po wpo 5062   Or wor 5063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-po 5064  df-so 5065

This theorem is referenced by:  sotric  5090  sotrieq  5091  soirri  5557  suppr  8418  infpr  8450  hartogslem1  8488  canth4  9507  canthwelem  9510  pwfseqlem4  9522  1ne0sr  9955  ltnr  10170  opsrtoslem2  19533  nodenselem4  31962  nodenselem5  31963  nodenselem7  31965  nolt02o  31970  noresle  31971  noprefixmo  31973  nosupbnd1lem1  31979  nosupbnd2lem1  31986  sltirr  31996  fin2solem  33525  fin2so  33526