Theorem dfss2 3263

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 3261	. . 3 ⊢ (A ⊆ B ↔ A = (A ∩ B))
2		dfcleq 2347	. . . 4 ⊢ (A = (A ∩ B) ↔ ∀x(x ∈ A ↔ x ∈ (A ∩ B)))
3		elin 3220	. . . . . 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 3209   ⊆ wss 3258

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  dfss3  3264  dfss2f  3265  ssel  3268  ssriv  3278  ssrdv  3279  sstr2  3280  eqss  3288  nss  3330  rabss2  3350  ssconb  3400  ssequn1  3434  unss  3438  ssin  3478  reldisj  3595  ssdif0  3610  difin0ss  3617  inssdif0  3618  ssundif  3634  sbcss  3661  sscon34  3662  pwss  3737  snss  3839  pwpw0  3856  pwsnALT  3883  disj5  3891  ssuni  3914  unissb  3922  intss  3948  iunss  4008  ssofss  4077  ssetkex  4295  dfpw2  4328  funimass4  5369  clos1induct  5881  dfnnc3  5886  ncssfin  6152