Theorem dfpss2 3355

Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
dfpss2	⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ A = B))

Proof of Theorem dfpss2

Step	Hyp	Ref	Expression
1		df-pss 3262	. 2 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ A ≠ B))
2		df-ne 2519	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
3	2	anbi2i 675	. 2 ⊢ ((A ⊆ B ∧ A ≠ B) ↔ (A ⊆ B ∧ ¬ A = B))
4	1, 3	bitri 240	1 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ A = B))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2517   ⊆ wss 3258   ⊊ wpss 3259

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

This proof depends on definitions:  df-bi 177  df-an 360  df-ne 2519  df-pss 3262

This theorem is used by:  dfpss3  3356  sspss  3369  psstr  3374  npss  3380  pssv  3591  disj4  3600  ssnelpss  3614  sfinltfin  4536