Theorem ssint 1980

Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse.

Assertion

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

Distinct variable group(s):   x,A   x,B

Proof of Theorem ssint

Step	Hyp	Ref	Expression
1		dfss3 1498	. 2 ⊢ (A ⊆ ∩B ↔ ∀y ∈ A y ∈ ∩B)
2		visset 1350	. . . 4 ⊢ y ∈ V
3	2	elint2 1972	. . 3 ⊢ (y ∈ ∩B ↔ ∀x ∈ B y ∈ x)
4	3	biral 1223	. 2 ⊢ (∀y ∈ A y ∈ ∩B ↔ ∀y ∈ A ∀x ∈ B y ∈ x)
5		ralcom 1312	. . 3 ⊢ (∀y ∈ A ∀x ∈ B y ∈ x ↔ ∀x ∈ B ∀y ∈ A y ∈ x)
6		dfss3 1498	. . . 4 ⊢ (A ⊆ x ↔ ∀y ∈ A y ∈ x)
7	6	biral 1223	. . 3 ⊢ (∀x ∈ B A ⊆ x ↔ ∀x ∈ B ∀y ∈ A y ∈ x)
8	5, 7	bitr4 154	. 2 ⊢ (∀y ∈ A ∀x ∈ B y ∈ x ↔ ∀x ∈ B A ⊆ x)
9	1, 4, 8	3bitr 155	1 ⊢ (A ⊆ ∩B ↔ ∀x ∈ B A ⊆ x)

Syntax hints:   ↔ wb 127   ∈ wel 803   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ∩cint 1965

This theorem is referenced by:  ssintub 1981  iinpw 2038  oneqmini 2272  fint 2769  iscard2 3660

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-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-int 1966

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