Theorem pssnssi 46389

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 4019	. . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
3	1, 2	mpbi 233	. 2 ⊢ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)
4	3	simpri 491	1 ⊢ ¬ 𝐵 ⊆ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ⊆ wss 3900   ⊊ wpss 4016

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 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ne 2960  df-ss 3901  df-pss 4017

This theorem is used by:  nsssmfmbf  48062