Theorem pssnssi 41737

Description: A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
pssnssi.1	⊢ 𝐴 ⊊ 𝐵

Assertion

Ref	Expression
pssnssi	⊢ ¬ 𝐵 ⊆ 𝐴

Proof of Theorem pssnssi

Step	Hyp	Ref	Expression
1		pssnssi.1	. . 3 ⊢ 𝐴 ⊊ 𝐵
2		dfpss3 4014	. . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
3	1, 2	mpbi 233	. 2 ⊢ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)
4	3	simpri 489	1 ⊢ ¬ 𝐵 ⊆ 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 399   ⊆ 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:  nsssmfmbf  43412