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Theorem fingch 10034
 Description: A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
fingch Fin ⊆ GCH

Proof of Theorem fingch
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssun1 4099 . 2 Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥𝑦𝑦 ≺ 𝒫 𝑥)})
2 df-gch 10032 . 2 GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥𝑦𝑦 ≺ 𝒫 𝑥)})
31, 2sseqtrri 3952 1 Fin ⊆ GCH
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1536  {cab 2776   ∪ cun 3879   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5030   ≺ csdm 8491  Fincfn 8492  GCHcgch 10031 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-gch 10032 This theorem is referenced by:  gch2  10086
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