Theorem dfpss3 4029

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 4028	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2		eqss 3937	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	baib 535	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ↔ 𝐵 ⊆ 𝐴))
4	3	notbid 318	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵 ⊆ 𝐴))
5	4	pm5.32i 574	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
6	1, 5	bitri 275	1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542   ⊆ wss 3889   ⊊ wpss 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2728  df-ne 2933  df-ss 3906  df-pss 3909

This theorem is referenced by:  pssirr  4043  pssn2lp  4044  ssnpss  4046  nsspssun  4208  pssdifcom1  4429  pssdifcom2  4430  php3  9143  fincssdom  10245  reclem2pr  10971  ressval3d  17216  islbs3  21153  ltslpss  27900  chpsscon3  31574  chpssati  32434  fundmpss  35949  lpssat  39459  lssat  39462  dihglblem6  41786  pssnssi  45531  mbfpsssmf  47211