Theorem ssint 3943

Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
ssint	⊢ (A ⊆ ∩B ↔ ∀x ∈ B A ⊆ x)

Distinct variable groups:   x,A   x,B

Proof of Theorem ssint

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss3 3264	. 2 ⊢ (A ⊆ ∩B ↔ ∀y ∈ A y ∈ ∩B)
2		vex 2863	. . . 4 ⊢ y ∈ V
3	2	elint2 3934	. . 3 ⊢ (y ∈ ∩B ↔ ∀x ∈ B y ∈ x)
4	3	ralbii 2639	. 2 ⊢ (∀y ∈ A y ∈ ∩B ↔ ∀y ∈ A ∀x ∈ B y ∈ x)
5		ralcom 2772	. . 3 ⊢ (∀y ∈ A ∀x ∈ B y ∈ x ↔ ∀x ∈ B ∀y ∈ A y ∈ x)
6		dfss3 3264	. . . 4 ⊢ (A ⊆ x ↔ ∀y ∈ A y ∈ x)
7	6	ralbii 2639	. . 3 ⊢ (∀x ∈ B A ⊆ x ↔ ∀x ∈ B ∀y ∈ A y ∈ x)
8	5, 7	bitr4i 243	. 2 ⊢ (∀y ∈ A ∀x ∈ B y ∈ x ↔ ∀x ∈ B A ⊆ x)
9	1, 4, 8	3bitri 262	1 ⊢ (A ⊆ ∩B ↔ ∀x ∈ B A ⊆ x)

This proof depends on syntax axioms:   ↔ wb 176   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258  ∩cint 3927

This proof depends on 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 proof 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-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is used by:  ssintab  3944  ssintub  3945  iinpw  4055  fint  5246