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Theorem bj-intabssel1 13790
Description: Version of intss1 3844 using a class abstraction and implicit substitution. Closed form of intmin3 3856. (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 13780 . 2 (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
5 intss1 3844 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6 33 1 (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wnf 1453  wcel 2141  {cab 2156  wnfc 2299  wss 3121   cint 3829
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 3830
This theorem is referenced by:  bj-omssind  13935
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