Theorem ssint 4895

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 3909	. 2 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵)
2		vex 3436	. . . 4 ⊢ 𝑦 ∈ V
3	2	elint2 4886	. . 3 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
4	3	ralbii 3092	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
5		ralcom 3166	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
6		dfss3 3909	. . . 4 ⊢ (𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
7	6	ralbii 3092	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
8	5, 7	bitr4i 277	. 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 205   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ∩ cint 4879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904  df-int 4880

This theorem is referenced by:  ssintab  4896  ssintub  4897  iinpw  5035  oneqmini  6317  fint  6653  sorpssint  7586  iscard2  9734  coftr  10029  isf32lem2  10110  inttsk  10530  dfrtrcl2  14773  isacs1i  17366  mrelatglb  18278  fbfinnfr  22992  fclscmp  23181  ssmxidllem  31641  fnssintima  33676  noextenddif  33871  eqscut2  34000  scutun12  34004  fneint  34537  topmeet  34553  igenval2  36224  ismrcd1  40520  dftrcl3  41328  dfrtrcl3  41341  sssalgen  43874  issalgend  43877  intubeu  46270  ipoglblem  46275