Theorem dfpss3 4021

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 4020	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2		eqss 3930	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	baib 540	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ↔ 𝐵 ⊆ 𝐴))
4	3	notbid 319	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵 ⊆ 𝐴))
5	4	pm5.32i 579	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
6	1, 5	bitri 276	1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   = wceq 1547   ⊆ wss 3883   ⊊ wpss 3884

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 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-ne 2935  df-ss 3900  df-pss 3903

This theorem is referenced by:  pssirr  4035  pssn2lp  4036  ssnpss  4038  nsspssun  4197  pssdifcom1  4418  pssdifcom2  4419  php3  9134  fincssdom  10237  reclem2pr  10963  ressval3d  17208  islbs3  21149  ltslpss  27919  chpsscon3  31593  chpssati  32453  fundmpss  36004  lpssat  39514  lssat  39517  dihglblem6  41841  pssnssi  45556  mbfpsssmf  47234