Theorem sspss 4044

Description: Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)

Assertion

Ref	Expression
sspss	⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem sspss

Step	Hyp	Ref	Expression
1		dfpss2 4030	. . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2	1	simplbi2 501	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 → 𝐴 ⊊ 𝐵))
3	2	con1d 145	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊊ 𝐵 → 𝐴 = 𝐵))
4	3	orrd 860	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
5		pssss 4040	. . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
6		eqimss 3986	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7	5, 6	jaoi 854	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)
8	4, 7	impbii 208	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∨ wo 844   = wceq 1540   ⊆ wss 3896   ⊊ wpss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-v 3442  df-in 3903  df-ss 3913  df-pss 3915

This theorem is referenced by:  sspsstri  4047  sspsstr  4050  psssstr  4051  ordsseleq  6317  sorpssuni  7626  sorpssint  7627  ssnnfi  9012  ssnnfiOLD  9013  ackbij1b  10074  fin23lem40  10186  zorng  10339  psslinpr  10866  suplem2pr  10888  ressval3d  17030  ressval3dOLD  17031  mrissmrcd  17423  pgpssslw  19292  pgpfac1lem5  19754  idnghm  23987  dfon2lem4  33889  finminlem  34577  lkrss2N  37408  dvh3dim3N  39689