Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-intabssel1 GIF version

Theorem bj-intabssel1 9929
 Description: Version of intss1 3630 using a class abstraction and implicit substitution. Closed form of intmin3 3642. (Contributed by BJ, 29-Nov-2019.)
Hypotheses
Ref Expression
bj-intabssel1.nf 𝑥𝐴
bj-intabssel1.nf2 𝑥𝜓
bj-intabssel1.is (𝑥 = 𝐴 → (𝜓𝜑))
Assertion
Ref Expression
bj-intabssel1 (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))

Proof of Theorem bj-intabssel1
StepHypRef Expression
1 bj-intabssel1.nf . . 3 𝑥𝐴
2 bj-intabssel1.nf2 . . 3 𝑥𝜓
3 bj-intabssel1.is . . 3 (𝑥 = 𝐴 → (𝜓𝜑))
41, 2, 3elabgf2 9919 . 2 (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
5 intss1 3630 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6 29 1 (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165   ⊆ wss 2917  ∩ cint 3615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-int 3616 This theorem is referenced by:  bj-omssind  10059
 Copyright terms: Public domain W3C validator