Theorem pssne 4047

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 3919	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
2	1	simprbi 503	1 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:   → wi 4   ≠ wne 2956   ⊆ wss 3899   ⊊ wpss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-pss 3919

This theorem is used by:  pssned  4049  pssirr  4051  canthp1lem2  10738  mrissmrcd  17814  xppss12  43283