Theorem sspss 4034

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 4020	. . . . 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 4030	. . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
6		eqimss 3977	. . 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 1539   ⊆ wss 3887   ⊊ wpss 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-in 3894  df-ss 3904  df-pss 3906

This theorem is referenced by:  sspsstri  4037  sspsstr  4040  psssstr  4041  ordsseleq  6295  sorpssuni  7585  sorpssint  7586  ssnnfi  8952  ssnnfiOLD  8953  ackbij1b  9995  fin23lem40  10107  zorng  10260  psslinpr  10787  suplem2pr  10809  ressval3d  16956  ressval3dOLD  16957  mrissmrcd  17349  pgpssslw  19219  pgpfac1lem5  19682  idnghm  23907  dfon2lem4  33762  finminlem  34507  lkrss2N  37183  dvh3dim3N  39463