Theorem dfpss3 4040

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 4039	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2		eqss 3949	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	baib 545	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ↔ 𝐵 ⊆ 𝐴))
4	3	notbid 321	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵 ⊆ 𝐴))
5	4	pm5.32i 585	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
6	1, 5	bitri 278	1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ⊆ wss 3902   ⊊ wpss 3903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-ne 2958  df-ss 3919  df-pss 3922

This theorem is used by:  pssirrOLD  4055  pssn2lp  4056  ssnpss  4058  nsspssun  4217  pssdifcom1  4448  pssdifcom2  4449  php3  9207  fincssdom  10329  reclem2pr  11061  ressval3d  17344  islbs3  21348  ltslpss  28181  chpsscon3  31992  chpssati  32852  fundmpss  36354  lpssat  39894  lssat  39897  dihglblem6  42221  pssnssi  45941  mbfpsssmf  47619