Theorem fingch 10667

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 4189	. 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
2		df-gch 10665	. 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
3	1, 2	sseqtrri 4034	1 ⊢ Fin ⊆ GCH

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1536  {cab 2713   ∪ cun 3962   ⊆ wss 3964  𝒫 cpw 4606   class class class wbr 5149   ≺ csdm 8989  Fincfn 8990  GCHcgch 10664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1541  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3969  df-ss 3981  df-gch 10665

This theorem is referenced by:  gch2  10719