Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fingch | Structured version Visualization version GIF version |
Description: A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
fingch | ⊢ Fin ⊆ GCH |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4072 | . 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) | |
2 | df-gch 10200 | . 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) | |
3 | 1, 2 | sseqtrri 3924 | 1 ⊢ Fin ⊆ GCH |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∀wal 1541 {cab 2714 ∪ cun 3851 ⊆ wss 3853 𝒫 cpw 4499 class class class wbr 5039 ≺ csdm 8603 Fincfn 8604 GCHcgch 10199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-un 3858 df-in 3860 df-ss 3870 df-gch 10200 |
This theorem is referenced by: gch2 10254 |
Copyright terms: Public domain | W3C validator |