Theorem ssint 4917

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 3926	. 2 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵)
2		vex 3442	. . . 4 ⊢ 𝑦 ∈ V
3	2	elint2 4906	. . 3 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
4	3	ralbii 3075	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
5		ralcom 3257	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
6		dfss3 3926	. . . 4 ⊢ (𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
7	6	ralbii 3075	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
8	5, 7	bitr4i 278	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥)
9	1, 4, 8	3bitri 297	1 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥)

Syntax hints:   ↔ wb 206   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3905  ∩ cint 4899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3440  df-ss 3922  df-int 4900

This theorem is referenced by:  ssintab  4918  ssintub  4919  iinpw  5058  oneqmini  6364  fint  6707  fnssintima  7303  sorpssint  7673  iscard2  9891  coftr  10186  isf32lem2  10267  inttsk  10687  dfrtrcl2  14987  isacs1i  17581  mrelatglb  18484  fbfinnfr  23744  fclscmp  23933  noextenddif  27596  eqscut2  27735  scutun12  27739  onsiso  28192  bdayn0p1  28281  ssdifidllem  33403  ssmxidllem  33420  fneint  36321  topmeet  36337  igenval2  38045  ismrcd1  42671  onintunirab  43200  dftrcl3  43693  dfrtrcl3  43706  sssalgen  46317  issalgend  46320  intubeu  48969  ipoglblem  48974