Theorem fingch 10735

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 4179	. 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
2		df-gch 10733	. 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
3	1, 2	sseqtrri 3962	1 ⊢ Fin ⊆ GCH

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401  ∀wal 1568  {cab 2741   ⊆ wss 3900   ∪ cun 4152  𝒫 cpw 4560   class class class wbr 5106   ≺ csdm 9000  Fincfn 9001  GCHcgch 10732

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-gch 10733

This theorem is used by:  gch2  10787