Theorem dfpss3 4066

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 4065	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2		eqss 3985	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	baib 538	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ↔ 𝐵 ⊆ 𝐴))
4	3	notbid 320	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵 ⊆ 𝐴))
5	4	pm5.32i 577	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
6	1, 5	bitri 277	1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1536   ⊆ wss 3939   ⊊ wpss 3940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3020  df-in 3946  df-ss 3955  df-pss 3957

This theorem is referenced by:  pssirr  4080  pssn2lp  4081  ssnpss  4083  nsspssun  4237  pssdifcom1  4438  pssdifcom2  4439  php3  8706  fincssdom  9748  reclem2pr  10473  ressval3d  16564  islbs3  19930  chpsscon3  29283  chpssati  30143  fundmpss  33013  lpssat  36153  lssat  36156  dihglblem6  38480  pssnssi  41373  mbfpsssmf  43066