Theorem dfss2 3262

Description: Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
dfss2	⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))

Distinct variable groups:   x,A   x,B

Proof of Theorem dfss2

Step	Hyp	Ref	Expression
1		dfss 3260	. . 3 ⊢ (A ⊆ B ↔ A = (A ∩ B))
2		dfcleq 2347	. . . 4 ⊢ (A = (A ∩ B) ↔ ∀x(x ∈ A ↔ x ∈ (A ∩ B)))
3		elin 3219	. . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
4	3	bibi2i 304	. . . . 5 ⊢ ((x ∈ A ↔ x ∈ (A ∩ B)) ↔ (x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
5	4	albii 1566	. . . 4 ⊢ (∀x(x ∈ A ↔ x ∈ (A ∩ B)) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
6	2, 5	bitri 240	. . 3 ⊢ (A = (A ∩ B) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
7	1, 6	bitri 240	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
8		pm4.71 611	. . 3 ⊢ ((x ∈ A → x ∈ B) ↔ (x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
9	8	albii 1566	. 2 ⊢ (∀x(x ∈ A → x ∈ B) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
10	7, 9	bitr4i 243	1 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∩ cin 3208   ⊆ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  dfss3  3263  dfss2f  3264  ssel  3267  ssriv  3277  ssrdv  3278  sstr2  3279  eqss  3287  nss  3329  rabss2  3349  ssconb  3399  ssequn1  3433  unss  3437  ssin  3477  reldisj  3594  ssdif0  3609  difin0ss  3616  inssdif0  3617  ssundif  3633  sbcss  3660  sscon34  3661  pwss  3736  snss  3838  pwpw0  3855  pwsnALT  3882  disj5  3890  ssuni  3913  unissb  3921  intss  3947  iunss  4007  ssofss  4076  ssetkex  4294  dfpw2  4327  funimass4  5368  clos1induct  5880  dfnnc3  5885  ncssfin  6151