Theorem ssnpss 4049

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 4032	. . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵))
2	1	simprbi 497	. 2 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵)
3	2	con2i 139	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3897   ⊊ wpss 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-v 3443  df-in 3904  df-ss 3914  df-pss 3916

This theorem is referenced by:  npss0  4391  sorpssuni  7639  sorpssint  7640  suplem2pr  10902  symgvalstruct  19092  symgvalstructOLD  19093  lsppratlem6  20512  atcvati  30977  finxpreclem3  35662  lsatcvat  37310