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Mirrors > Home > MPE Home > Th. List > elgch | Structured version Visualization version GIF version |
Description: Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
elgch | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-gch 10200 | . . . 4 ⊢ GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) | |
2 | 1 | eleq2i 2822 | . . 3 ⊢ (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)})) |
3 | elun 4049 | . . 3 ⊢ (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)})) | |
4 | 2, 3 | bitri 278 | . 2 ⊢ (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)})) |
5 | breq1 5042 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥)) | |
6 | pweq 4515 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴) | |
7 | 6 | breq2d 5051 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦 ↔ 𝑥 ≺ 𝒫 𝐴)) |
8 | 5, 7 | anbi12d 634 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))) |
9 | 8 | notbid 321 | . . . . 5 ⊢ (𝑦 = 𝐴 → (¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))) |
10 | 9 | albidv 1928 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))) |
11 | 10 | elabg 3574 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))) |
12 | 11 | orbi2d 916 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) |
13 | 4, 12 | syl5bb 286 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∀wal 1541 = wceq 1543 ∈ wcel 2112 {cab 2714 ∪ cun 3851 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-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-gch 10200 |
This theorem is referenced by: gchi 10203 engch 10207 hargch 10252 alephgch 10253 |
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