Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8633 | . . . . . . 7 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5605 | . . . . . 6 ⊢ (𝐵 ≺ 𝒫 𝐴 → 𝐵 ∈ V) |
3 | 2 | adantl 485 | . . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V) |
4 | breq2 5057 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝐵)) | |
5 | breq1 5056 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝐵 ≺ 𝒫 𝐴)) | |
6 | 4, 5 | anbi12d 634 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴))) |
7 | 6 | spcegv 3512 | . . . . 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 1788 | . . . 4 ⊢ (∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)) | |
10 | 8, 9 | sylib 221 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)) |
11 | elgch 10236 | . . . . . 6 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) | |
12 | 11 | ibi 270 | . . . . 5 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))) |
13 | 12 | orcomd 871 | . . . 4 ⊢ (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin)) |
14 | 13 | ord 864 | . . 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 1118 | 1 ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2110 Vcvv 3408 𝒫 cpw 4513 class class class wbr 5053 ≺ csdm 8625 Fincfn 8626 GCHcgch 10234 |
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 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
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 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-dom 8628 df-sdom 8629 df-gch 10235 |
This theorem is referenced by: gchen1 10239 gchen2 10240 gchpwdom 10284 gchaleph 10285 |
Copyright terms: Public domain | W3C validator |