Theorem intminss 4503

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 3363	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
3		intss1 4492	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)
4	2, 3	sylbir 225	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916   ⊆ wss 3574  ∩ cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by:  onintss  5775  knatar  6607  cardonle  8783  coftr  9095  wuncss  9567  ist1-3  21153  sigagenss  30212  ldgenpisyslem1  30226  dynkin  30230  fneint  32343  igenmin  33863  pclclN  35177  dfrcl2  37966