Theorem ssint 4925

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 3928	. 2 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵)
2		vex 3461	. . . 4 ⊢ 𝑦 ∈ V
3	2	elint2 4915	. . 3 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
4	3	ralbii 3111	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
5		ralcom 3293	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
6		dfss3 3928	. . . 4 ⊢ (𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
7	6	ralbii 3111	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥)
8	5, 7	bitr4i 281	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥)
9	1, 4, 8	3bitri 300	1 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥)

Syntax hints:   ↔ wb 209   ∈ wcel 2145  ∀wral 3079   ⊆ wss 3907  ∩ cint 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-v 3459  df-ss 3924  df-int 4909

This theorem is referenced by:  ssintab  4926  ssintub  4927  iinpw  5068  oneqmini  6403  fint  6747  fnssintima  7350  sorpssint  7720  iscard2  9950  coftr  10245  isf32lem2  10326  inttsk  10747  dfrtrcl2  15089  isacs1i  17703  mrelatglb  18606  ssdifidllem  21444  fbfinnfr  23959  fclscmp  24148  noextenddif  27790  eqcuts2  27937  cutsun12  27941  oniso  28422  bdayn0p1  28520  ssmxidllem  33673  fneint  36721  topmeet  36737  igenval2  38577  ismrcd1  43291  onintunirab  43816  dftrcl3  44308  dfrtrcl3  44321  sssalgen  46907  issalgend  46910  intubeu  49613  ipoglblem  49618