Theorem fingch 10546

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 4132	. 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
2		df-gch 10544	. 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
3	1, 2	sseqtrri 3985	1 ⊢ Fin ⊆ GCH

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1540  {cab 2715   ∪ cun 3901   ⊆ wss 3903  𝒫 cpw 4556   class class class wbr 5100   ≺ csdm 8894  Fincfn 8895  GCHcgch 10543

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-gch 10544

This theorem is referenced by:  gch2  10598