Theorem pssned 4042

Description: Proper subclasses are unequal. Deduction form of pssne 4040. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
pssssd.1	⊢ (𝜑 → 𝐴 ⊊ 𝐵)

Assertion

Ref	Expression
pssned	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Proof of Theorem pssned

Step	Hyp	Ref	Expression
1		pssssd.1	. 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵)
2		pssne 4040	. 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   ≠ wne 2933   ⊊ wpss 3891

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 207  df-an 396  df-pss 3910

This theorem is referenced by:  omsucne  7829  ackbij1lem15  10146  canthnumlem  10562  canthp1lem2  10567  mrieqv2d  17596  slwpss  19578  hashpss  32897  topdifinffinlem  37677  lsatssn0  39462  islshpcv  39513  lkrpssN  39623