Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > gchor | Structured version Visualization version GIF version |
Description: If 𝐴 ≤ 𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then either 𝐴 = 𝐵 or 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gchor | ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → (𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 796 | . . 3 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≼ 𝒫 𝐴) | |
2 | brdom2 7985 | . . 3 ⊢ (𝐵 ≼ 𝒫 𝐴 ↔ (𝐵 ≺ 𝒫 𝐴 ∨ 𝐵 ≈ 𝒫 𝐴)) | |
3 | 1, 2 | sylib 208 | . 2 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → (𝐵 ≺ 𝒫 𝐴 ∨ 𝐵 ≈ 𝒫 𝐴)) |
4 | gchen1 9447 | . . . . 5 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)) → 𝐴 ≈ 𝐵) | |
5 | 4 | expr 643 | . . . 4 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐴 ≼ 𝐵) → (𝐵 ≺ 𝒫 𝐴 → 𝐴 ≈ 𝐵)) |
6 | 5 | adantrr 753 | . . 3 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → (𝐵 ≺ 𝒫 𝐴 → 𝐴 ≈ 𝐵)) |
7 | 6 | orim1d 884 | . 2 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → ((𝐵 ≺ 𝒫 𝐴 ∨ 𝐵 ≈ 𝒫 𝐴) → (𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴))) |
8 | 3, 7 | mpd 15 | 1 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → (𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 ∈ wcel 1990 𝒫 cpw 4158 class class class wbr 4653 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 Fincfn 7955 GCHcgch 9442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-f1o 5895 df-en 7956 df-dom 7957 df-sdom 7958 df-gch 9443 |
This theorem is referenced by: gchdomtri 9451 gchpwdom 9492 |
Copyright terms: Public domain | W3C validator |