Theorem pssnssi 45852

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 4046	. . 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 3908   ⊊ wpss 3909

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 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2758  df-ne 2962  df-ss 3925  df-pss 3928

This theorem is used by:  nsssmfmbf  47526