Theorem sspss 4055

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 4041	. . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2	1	simplbi2 500	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 → 𝐴 ⊊ 𝐵))
3	2	con1d 145	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊊ 𝐵 → 𝐴 = 𝐵))
4	3	orrd 864	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
5		pssss 4051	. . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
6		eqimss 3993	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7	5, 6	jaoi 858	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)
8	4, 7	impbii 209	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 848   = wceq 1542   ⊆ wss 3902   ⊊ wpss 3903

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-cleq 2729  df-ne 2934  df-ss 3919  df-pss 3922

This theorem is referenced by:  sspsstri  4058  sspsstr  4061  psssstr  4062  ordsseleq  6347  sorpssuni  7679  sorpssint  7680  ssnnfi  9098  ackbij1b  10152  fin23lem40  10265  zorng  10418  psslinpr  10946  suplem2pr  10968  ressval3d  17177  mrissmrcd  17567  pgpssslw  19547  pgpfac1lem5  20014  idnghm  24691  slelss  27891  dfon2lem4  35959  finminlem  36493  lkrss2N  39466  dvh3dim3N  41746  ordsssucb  43613