Theorem sspss 1569

Description: Subclass in terms of proper subclass.

Assertion

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

Proof of Theorem sspss

Step	Hyp	Ref	Expression
1		dfpss2 1557	. . . . . 6 ⊢ (A ⊂ B ↔ (A ⊆ B ∧ ¬ A = B))
2	1	biimpr 134	. . . . 5 ⊢ ((A ⊆ B ∧ ¬ A = B) → A ⊂ B)
3	2	exp 291	. . . 4 ⊢ (A ⊆ B → (¬ A = B → A ⊂ B))
4	3	con1d 85	. . 3 ⊢ (A ⊆ B → (¬ A ⊂ B → A = B))
5	4	orrd 203	. 2 ⊢ (A ⊆ B → (A ⊂ B ∨ A = B))
6		pssss 1567	. . 3 ⊢ (A ⊂ B → A ⊆ B)
7		eqimss 1548	. . 3 ⊢ (A = B → A ⊆ B)
8	6, 7	jaoi 275	. 2 ⊢ ((A ⊂ B ∨ A = B) → A ⊆ B)
9	5, 8	impbi 139	1 ⊢ (A ⊆ B ↔ (A ⊂ B ∨ A = B))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ⊆ wss 1487   ⊂ wpss 1488

This theorem is referenced by:  sspsstri 1572  ssnpss 1573  sspsstr 1575  psssstr 1576  ssnn 3429  zorn2 3612  psslinpr 3929  suplem2pr 3956

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-in 1491  df-ss 1492  df-pss 1494

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