Theorem bj-intabssel1 11963

Description: Version of intss1 3709 using a class abstraction and implicit substitution. Closed form of intmin3 3721. (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 11953	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
5		intss1 3709	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)
6	4, 5	syl6 33	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1290  Ⅎwnf 1395   ∈ wcel 1439  {cab 2075  Ⅎwnfc 2216   ⊆ wss 3000  ∩ cint 3694

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-int 3695

This theorem is referenced by:  bj-omssind  12103