Theorem ssnpss 4031

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 4014	. . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵))
2	1	simprbi 500	. 2 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵)
3	2	con2i 141	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3881   ⊊ wpss 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-in 3888  df-ss 3898  df-pss 3900

This theorem is referenced by:  npss0  4353  sorpssuni  7438  sorpssint  7439  suplem2pr  10464  symgvalstruct  18517  lsppratlem6  19917  atcvati  30169  finxpreclem3  34810  lsatcvat  36346