Theorem intminss 3899

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 2920	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
3		intss1 3889	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)
4	2, 3	sylbir 135	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {crab 2479   ⊆ wss 3157  ∩ cint 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-in 3163  df-ss 3170  df-int 3875

This theorem is referenced by:  onintss  4425  cardonle  7254