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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 3738 | . 2 ⊢ Fin ⊆ (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) | |
2 | df-gch 9322 | . 2 ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) | |
3 | 1, 2 | sseqtr4i 3601 | 1 ⊢ Fin ⊆ GCH |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ∀wal 1473 {cab 2596 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 ≺ csdm 7840 Fincfn 7841 GCHcgch 9321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-gch 9322 |
This theorem is referenced by: gch2 9376 |
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