Theorem ssnpss 4055

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 4037	. . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵))
2	1	simprbi 503	. 2 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵)
3	2	con2i 140	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ⊆ wss 3899   ⊊ wpss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-ne 2957  df-ss 3916  df-pss 3919

This theorem is used by:  npss0  4361  nvpss  4363  sorpssuni  7737  sorpssint  7738  suplem2pr  11119  symgvalstruct  19591  lsppratlem6  21410  atcvati  32970  finxpreclem3  38284  lsatcvat  40075