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Theorem superficl 42308
Description: The class of all supersets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)
Hypothesis
Ref Expression
superficl.a 𝐴 = {𝑧𝐵𝑧}
Assertion
Ref Expression
superficl 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴   𝑧,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem superficl
StepHypRef Expression
1 superficl.a . 2 𝐴 = {𝑧𝐵𝑧}
2 vex 3478 . . 3 𝑥 ∈ V
32inex1 5317 . 2 (𝑥𝑦) ∈ V
4 sseq2 4008 . 2 (𝑧 = (𝑥𝑦) → (𝐵𝑧𝐵 ⊆ (𝑥𝑦)))
5 sseq2 4008 . 2 (𝑧 = 𝑥 → (𝐵𝑧𝐵𝑥))
6 sseq2 4008 . 2 (𝑧 = 𝑦 → (𝐵𝑧𝐵𝑦))
7 ssin 4230 . . 3 ((𝐵𝑥𝐵𝑦) ↔ 𝐵 ⊆ (𝑥𝑦))
87biimpi 215 . 2 ((𝐵𝑥𝐵𝑦) → 𝐵 ⊆ (𝑥𝑦))
91, 3, 4, 5, 6, 8cllem0 42307 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  {cab 2709  wral 3061  Vcvv 3474  cin 3947  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-v 3476  df-in 3955  df-ss 3965
This theorem is referenced by: (None)
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