Theorem pssne 4031

Description: Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)

Assertion

Ref	Expression
pssne	⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)

Proof of Theorem pssne

Step	Hyp	Ref	Expression
1		df-pss 3906	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
2	1	simprbi 497	1 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   ≠ wne 2943   ⊆ wss 3887   ⊊ wpss 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-pss 3906

This theorem is referenced by:  pssned  4033  canthp1lem2  10409  mrissmrcd  17349  xppss12  40204