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Theorem bj-intabssel1 11336
Description: Version of intss1 3698 using a class abstraction and implicit substitution. Closed form of intmin3 3710. (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 11326 . 2 (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
5 intss1 3698 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6 33 1 (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wnf 1394  wcel 1438  {cab 2074  wnfc 2215  wss 2997   cint 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-int 3684
This theorem is referenced by:  bj-omssind  11476
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