Theorem intminss 4911

Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)

Hypothesis

Ref	Expression
intminss.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
intminss	⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem intminss

Step	Hyp	Ref	Expression
1		intminss.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	elrab 3636	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
3		intss1 4900	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)
4	2, 3	sylbir 236	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392   ⊆ wss 3890  ∩ cint 4884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-ss 3907  df-int 4885

This theorem is referenced by:  onintss  6369  knatar  7308  dfttrcl2  9643  cardonle  9879  coftr  10193  wuncss  10666  ist1-3  23339  sigagenss  34340  ldgenpisyslem1  34354  dynkin  34358  tz9.1regs  35322  fneint  36583  igenmin  38438  pclclN  40390  dfrcl2  44125