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Theorem fingch 9646
 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 3925 . 2 Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥𝑦𝑦 ≺ 𝒫 𝑥)})
2 df-gch 9644 . 2 GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥𝑦𝑦 ≺ 𝒫 𝑥)})
31, 2sseqtr4i 3785 1 Fin ⊆ GCH
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 382  ∀wal 1628  {cab 2756   ∪ cun 3719   ⊆ wss 3721  𝒫 cpw 4295   class class class wbr 4784   ≺ csdm 8107  Fincfn 8108  GCHcgch 9643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-in 3728  df-ss 3735  df-gch 9644 This theorem is referenced by:  gch2  9698
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