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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 4147 | . 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) | |
2 | df-gch 10037 | . 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) | |
3 | 1, 2 | sseqtrri 4003 | 1 ⊢ Fin ⊆ GCH |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∀wal 1531 {cab 2799 ∪ cun 3933 ⊆ wss 3935 𝒫 cpw 4538 class class class wbr 5058 ≺ csdm 8502 Fincfn 8503 GCHcgch 10036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-in 3942 df-ss 3951 df-gch 10037 |
This theorem is referenced by: gch2 10091 |
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