Theorem pssne 4054

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 3926	. 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 2960   ⊆ wss 3906   ⊊ wpss 3907

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 3926

This theorem is used by:  pssned  4056  pssirr  4058  canthp1lem2  10655  mrissmrcd  17720  xppss12  43060