Theorem bj-intabssel1 13671

Description: Version of intss1 3839 using a class abstraction and implicit substitution. Closed form of intmin3 3851. (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 13661	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
5		intss1 3839	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)
6	4, 5	syl6 33	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295   ⊆ wss 3116  ∩ cint 3824

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-int 3825

This theorem is referenced by:  bj-omssind  13817