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Theorem ssficl 40687
 Description: The class of all subsets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)
Hypothesis
Ref Expression
ssficl.a 𝐴 = {𝑧𝑧𝐵}
Assertion
Ref Expression
ssficl 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 ssficl
StepHypRef Expression
1 ssficl.a . 2 𝐴 = {𝑧𝑧𝐵}
2 vex 3414 . . 3 𝑥 ∈ V
32inex1 5192 . 2 (𝑥𝑦) ∈ V
4 sseq1 3920 . 2 (𝑧 = (𝑥𝑦) → (𝑧𝐵 ↔ (𝑥𝑦) ⊆ 𝐵))
5 sseq1 3920 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
6 sseq1 3920 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
7 ssinss1 4145 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
87adantr 484 . 2 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ⊆ 𝐵)
91, 3, 4, 5, 6, 8cllem0 40684 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539   ∈ wcel 2112  {cab 2736  ∀wral 3071  Vcvv 3410   ∩ cin 3860   ⊆ wss 3861 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-v 3412  df-in 3868  df-ss 3878 This theorem is referenced by: (None)
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