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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 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 11953 . 2 (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
5 intss1 3709 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6 33 1 (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1290  wnf 1395  wcel 1439  {cab 2075  wnfc 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
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