Theorem ssint 4969

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 3984	. 2 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵)
2		vex 3482	. . . 4 ⊢ 𝑦 ∈ V
3	2	elint2 4958	. . 3 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
4	3	ralbii 3091	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
5		ralcom 3287	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
6		dfss3 3984	. . . 4 ⊢ (𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
7	6	ralbii 3091	. . 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 2106  ∀wral 3059   ⊆ wss 3963  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-int 4952

This theorem is referenced by:  ssintab  4970  ssintub  4971  iinpw  5111  oneqmini  6438  fint  6788  fnssintima  7382  sorpssint  7752  iscard2  10014  coftr  10311  isf32lem2  10392  inttsk  10812  dfrtrcl2  15098  isacs1i  17702  mrelatglb  18618  fbfinnfr  23865  fclscmp  24054  noextenddif  27728  eqscut2  27866  scutun12  27870  ssdifidllem  33464  ssmxidllem  33481  fneint  36331  topmeet  36347  igenval2  38053  ismrcd1  42686  onintunirab  43216  dftrcl3  43710  dfrtrcl3  43723  sssalgen  46291  issalgend  46294  intubeu  48773  ipoglblem  48778