Theorem pssnssi 39283

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 3693	. . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
3	1, 2	mpbi 220	. 2 ⊢ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)
4	3	simpri 478	1 ⊢ ¬ 𝐵 ⊆ 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 384   ⊆ wss 3574   ⊊ wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  nsssmfmbf  40987