Theorem sspss 4056

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 4042	. . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2	1	simplbi2 504	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 → 𝐴 ⊊ 𝐵))
3	2	con1d 145	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊊ 𝐵 → 𝐴 = 𝐵))
4	3	orrd 874	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
5		pssss 4052	. . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
6		eqimss 3995	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7	5, 6	jaoi 868	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)
8	4, 7	impbii 211	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∨ wo 858   = wceq 1561   ⊆ wss 3905   ⊊ wpss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1801  df-cleq 2755  df-ne 2959  df-ss 3922  df-pss 3925

This theorem is referenced by:  sspsstri  4060  sspsstr  4063  psssstr  4064  ordsseleq  6376  sorpssuni  7716  sorpssint  7717  ssnnfi  9139  ackbij1b  10195  fin23lem40  10309  zorng  10462  psslinpr  10990  suplem2pr  11012  ressval3d  17283  mrissmrcd  17673  pgpssslw  19655  pgpfac1lem5  20122  idnghm  24804  leslss  28003  dfon2lem4  36135  finminlem  36679  lkrss2N  39794  dvh3dim3N  42074  ordsssucb  43913