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Mirrors > Home > MPE Home > Th. List > gchen1 | Structured version Visualization version GIF version |
Description: If 𝐴 ≤ 𝐵 < 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐴 = 𝐵 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gchen1 | ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 794 | . 2 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)) → 𝐴 ≼ 𝐵) | |
2 | gchi 9446 | . . . . . . 7 ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin) | |
3 | 2 | 3com23 1271 | . . . . . 6 ⊢ ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴 ∧ 𝐴 ≺ 𝐵) → 𝐴 ∈ Fin) |
4 | 3 | 3expia 1267 | . . . . 5 ⊢ ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) → (𝐴 ≺ 𝐵 → 𝐴 ∈ Fin)) |
5 | 4 | con3dimp 457 | . . . 4 ⊢ (((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ≺ 𝐵) |
6 | 5 | an32s 846 | . . 3 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ 𝐴 ≺ 𝐵) |
7 | 6 | adantrl 752 | . 2 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)) → ¬ 𝐴 ≺ 𝐵) |
8 | bren2 7986 | . 2 ⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) | |
9 | 1, 7, 8 | sylanbrc 698 | 1 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ 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: gchor 9449 gchcda1 9478 gchcdaidm 9490 gchxpidm 9491 gchhar 9501 |
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