Theorem fingch 10629

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 4151	. 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
2		df-gch 10627	. 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
3	1, 2	sseqtrri 4006	1 ⊢ Fin ⊆ GCH

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1537  {cab 2712   ∪ cun 3922   ⊆ wss 3924  𝒫 cpw 4573   class class class wbr 5116   ≺ csdm 8952  Fincfn 8953  GCHcgch 10626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3459  df-un 3929  df-ss 3941  df-gch 10627

This theorem is referenced by:  gch2  10681