Theorem ssnpss 4079

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 4062	. . 3 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵))
2	1	simprbi 499	. 2 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵)
3	2	con2i 141	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3935   ⊊ wpss 3936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-in 3942  df-ss 3951  df-pss 3953

This theorem is referenced by:  npss0  4396  sorpssuni  7452  sorpssint  7453  suplem2pr  10469  symgvalstruct  18519  lsppratlem6  19918  atcvati  30157  finxpreclem3  34668  lsatcvat  36180