Theorem dfpss3 3356

Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
dfpss3	⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ B ⊆ A))

Proof of Theorem dfpss3

Step	Hyp	Ref	Expression
1		dfpss2 3355	. 2 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ A = B))
2		eqss 3288	. . . . 5 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
3	2	baib 871	. . . 4 ⊢ (A ⊆ B → (A = B ↔ B ⊆ A))
4	3	notbid 285	. . 3 ⊢ (A ⊆ B → (¬ A = B ↔ ¬ B ⊆ A))
5	4	pm5.32i 618	. 2 ⊢ ((A ⊆ B ∧ ¬ A = B) ↔ (A ⊆ B ∧ ¬ B ⊆ A))
6	1, 5	bitri 240	1 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ B ⊆ A))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3258   ⊊ wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by:  pssirr  3370  pssn2lp  3371  ssnpss  3373  nsspssun  3489  npss0  3590  pssdifcom1  3636  pssdifcom2  3637  dfpss4  3889