Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssficl Structured version   Visualization version   GIF version

Theorem ssficl 43559
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 3482 . . 3 𝑥 ∈ V
32inex1 5323 . 2 (𝑥𝑦) ∈ V
4 sseq1 4021 . 2 (𝑧 = (𝑥𝑦) → (𝑧𝐵 ↔ (𝑥𝑦) ⊆ 𝐵))
5 sseq1 4021 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
6 sseq1 4021 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
7 ssinss1 4254 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
87adantr 480 . 2 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ⊆ 𝐵)
91, 3, 4, 5, 6, 8cllem0 43556 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  cin 3962  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-in 3970  df-ss 3980
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator