Theorem bj-intabssel1 13825

Description: Version of intss1 3846 using a class abstraction and implicit substitution. Closed form of intmin3 3858. (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 13815	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
5		intss1 3846	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)
6	4, 5	syl6 33	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1348  Ⅎwnf 1453   ∈ wcel 2141  {cab 2156  Ⅎwnfc 2299   ⊆ wss 3121  ∩ cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-int 3832

This theorem is referenced by:  bj-omssind  13970