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Theorem bj-intabssel1 10316
Description: Version of intss1 3658 using a class abstraction and implicit substitution. Closed form of intmin3 3670. (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 10306 . 2 (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
5 intss1 3658 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6 33 1 (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wnf 1365  wcel 1409  {cab 2042  wnfc 2181  wss 2945   cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-int 3644
This theorem is referenced by:  bj-omssind  10446
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