Theorem pssned 4031

Description: Proper subclasses are unequal. Deduction form of pssne 4029. (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 4029	. 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:   → wi 4   ≠ wne 2959   ⊊ wpss 4016

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 4017

This theorem is used by:  omsucne  7928  ackbij1lem15  10348  canthnumlem  10770  canthp1lem2  10775  hashpss  14593  mrieqv2d  17854  slwpss  19868  topdifinffinlem  38531  lsatssn0  40318  islshpcv  40369  lkrpssN  40479