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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 8367 | . . . . . . 7 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5497 | . . . . . 6 ⊢ (𝐵 ≺ 𝒫 𝐴 → 𝐵 ∈ V) |
3 | 2 | adantl 482 | . . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V) |
4 | breq2 4968 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝐵)) | |
5 | breq1 4967 | . . . . . . 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 3538 | . . . . 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 1763 | . . . 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 9893 | . . . . . 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 866 | . . . 4 ⊢ (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin)) |
14 | 13 | ord 859 | . . 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 1109 | 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 842 ∧ w3a 1080 ∀wal 1520 = wceq 1522 ∃wex 1762 ∈ wcel 2080 Vcvv 3436 𝒫 cpw 4455 class class class wbr 4964 ≺ csdm 8359 Fincfn 8360 GCHcgch 9891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-br 4965 df-opab 5027 df-xp 5452 df-rel 5453 df-dom 8362 df-sdom 8363 df-gch 9892 |
This theorem is referenced by: gchen1 9896 gchen2 9897 gchpwdom 9941 gchaleph 9942 |
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