Theorem bj-intabssel1 13168

Description: Version of intss1 3794 using a class abstraction and implicit substitution. Closed form of intmin3 3806. (Contributed by BJ, 29-Nov-2019.)

Hypotheses

Ref	Expression
bj-intabssel1.nf	⊢ Ⅎ𝑥𝐴
bj-intabssel1.nf2	⊢ Ⅎ𝑥𝜓
bj-intabssel1.is	⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))

Assertion

Ref	Expression
bj-intabssel1	⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Proof of Theorem bj-intabssel1

Step	Hyp	Ref	Expression
1		bj-intabssel1.nf	. . 3 ⊢ Ⅎ𝑥𝐴
2		bj-intabssel1.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-intabssel1.is	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
4	1, 2, 3	elabgf2 13158	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
5		intss1 3794	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)
6	4, 5	syl6 33	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1332  Ⅎwnf 1437   ∈ wcel 1481  {cab 2126  Ⅎwnfc 2269   ⊆ wss 3076  ∩ cint 3779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-int 3780

This theorem is referenced by:  bj-omssind  13304