Theorem dfpss3 3977

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 3976	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2		eqss 3892	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	baib 539	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ↔ 𝐵 ⊆ 𝐴))
4	3	notbid 321	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵 ⊆ 𝐴))
5	4	pm5.32i 578	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
6	1, 5	bitri 278	1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1542   ⊆ wss 3843   ⊊ wpss 3844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3400  df-in 3850  df-ss 3860  df-pss 3862

This theorem is referenced by:  pssirr  3991  pssn2lp  3992  ssnpss  3994  nsspssun  4148  pssdifcom1  4376  pssdifcom2  4377  php3  8753  fincssdom  9823  reclem2pr  10548  ressval3d  16664  islbs3  20046  chpsscon3  29438  chpssati  30298  fundmpss  33312  sltlpss  33725  lpssat  36650  lssat  36653  dihglblem6  38977  pssnssi  42189  mbfpsssmf  43857