Theorem intminss 4944

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 3653	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
3		intss1 4933	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)
4	2, 3	sylbir 238	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3419   ⊆ wss 3908  ∩ cint 4917

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-ss 3925  df-int 4918

This theorem is used by:  onintss  6420  knatar  7368  dfttrcl2  9703  cardonle  9962  coftr  10275  wuncss  10748  ist1-3  23543  sigagenss  34571  ldgenpisyslem1  34585  dynkin  34589  tz9.1regs  35571  fneint  36900  igenmin  38756  pclclN  40706  dfrcl2  44441