Theorem ssnpss 4059

Description: Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ssnpss	⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)

Proof of Theorem ssnpss

Step	Hyp	Ref	Expression
1		dfpss3 4042	. . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵))
2	1	simprbi 496	. 2 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵)
3	2	con2i 139	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3905   ⊊ wpss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-ne 2926  df-ss 3922  df-pss 3925

This theorem is referenced by:  npss0  4401  sorpssuni  7672  sorpssint  7673  suplem2pr  10966  symgvalstruct  19294  lsppratlem6  21077  atcvati  32348  finxpreclem3  37369  lsatcvat  39031