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Mirrors > Home > MPE Home > Th. List > gchi | Structured version Visualization version GIF version |
Description: The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gchi | ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8504 | . . . . . . 7 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5601 | . . . . . 6 ⊢ (𝐵 ≺ 𝒫 𝐴 → 𝐵 ∈ V) |
3 | 2 | adantl 482 | . . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V) |
4 | breq2 5061 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝐵)) | |
5 | breq1 5060 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝐵 ≺ 𝒫 𝐴)) | |
6 | 4, 5 | anbi12d 630 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴))) |
7 | 6 | spcegv 3594 | . . . . 5 ⊢ (𝐵 ∈ V → ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))) |
8 | 3, 7 | mpcom 38 | . . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)) |
9 | df-ex 1772 | . . . 4 ⊢ (∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)) | |
10 | 8, 9 | sylib 219 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)) |
11 | elgch 10032 | . . . . . 6 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) | |
12 | 11 | ibi 268 | . . . . 5 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))) |
13 | 12 | orcomd 865 | . . . 4 ⊢ (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin)) |
14 | 13 | ord 858 | . . 3 ⊢ (𝐴 ∈ GCH → (¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)) |
15 | 10, 14 | syl5 34 | . 2 ⊢ (𝐴 ∈ GCH → ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)) |
16 | 15 | 3impib 1108 | 1 ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 𝒫 cpw 4535 class class class wbr 5057 ≺ csdm 8496 Fincfn 8497 GCHcgch 10030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dom 8499 df-sdom 8500 df-gch 10031 |
This theorem is referenced by: gchen1 10035 gchen2 10036 gchpwdom 10080 gchaleph 10081 |
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