Theorem sspss 4099

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 4085	. . . . 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 861	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
5		pssss 4095	. . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
6		eqimss 4040	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7	5, 6	jaoi 855	. 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 845   = wceq 1541   ⊆ wss 3948   ⊊ wpss 3949

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-in 3955  df-ss 3965  df-pss 3967

This theorem is referenced by:  sspsstri  4102  sspsstr  4105  psssstr  4106  ordsseleq  6393  sorpssuni  7721  sorpssint  7722  ssnnfi  9168  ssnnfiOLD  9169  ackbij1b  10233  fin23lem40  10345  zorng  10498  psslinpr  11025  suplem2pr  11047  ressval3d  17190  ressval3dOLD  17191  mrissmrcd  17583  pgpssslw  19481  pgpfac1lem5  19948  idnghm  24259  dfon2lem4  34753  finminlem  35198  lkrss2N  38034  dvh3dim3N  40315  ordsssucb  42075