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Theorem superficl 39917
 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 3496 . . 3 𝑥 ∈ V
32inex1 5212 . 2 (𝑥𝑦) ∈ V
4 sseq2 3991 . 2 (𝑧 = (𝑥𝑦) → (𝐵𝑧𝐵 ⊆ (𝑥𝑦)))
5 sseq2 3991 . 2 (𝑧 = 𝑥 → (𝐵𝑧𝐵𝑥))
6 sseq2 3991 . 2 (𝑧 = 𝑦 → (𝐵𝑧𝐵𝑦))
7 ssin 4205 . . 3 ((𝐵𝑥𝐵𝑦) ↔ 𝐵 ⊆ (𝑥𝑦))
87biimpi 218 . 2 ((𝐵𝑥𝐵𝑦) → 𝐵 ⊆ (𝑥𝑦))
91, 3, 4, 5, 6, 8cllem0 39916 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 398   = wceq 1531   ∈ wcel 2108  {cab 2797  ∀wral 3136  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495  df-in 3941  df-ss 3950 This theorem is referenced by: (None)
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