Theorem fingch 10615

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.

Step	Hyp	Ref	Expression
1		ssun1 4172	. 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
2		df-gch 10613	. 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
3	1, 2	sseqtrri 4019	1 ⊢ Fin ⊆ GCH

Syntax hints:  ¬ wn 3   ∧ wa 397  ∀wal 1540  {cab 2710   ∪ cun 3946   ⊆ wss 3948  𝒫 cpw 4602   class class class wbr 5148   ≺ csdm 8935  Fincfn 8936  GCHcgch 10612

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3953  df-in 3955  df-ss 3965  df-gch 10613

This theorem is referenced by:  gch2  10667