Theorem dfpss2 3354

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 3261	. 2 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ A ≠ B))
2		df-ne 2518	. . 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))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2516   ⊆ wss 3257   ⊊ wpss 3258

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 177  df-an 360  df-ne 2518  df-pss 3261

This theorem is referenced by:  dfpss3  3355  sspss  3368  psstr  3373  npss  3379  pssv  3590  disj4  3599  ssnelpss  3613  sfinltfin  4535