![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alephsucdom | Structured version Visualization version GIF version |
Description: A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
Ref | Expression |
---|---|
alephsucdom | ⊢ (𝐵 ∈ On → (𝐴 ≼ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘suc 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephordilem1 9484 | . . 3 ⊢ (𝐵 ∈ On → (ℵ‘𝐵) ≺ (ℵ‘suc 𝐵)) | |
2 | domsdomtr 8636 | . . . 4 ⊢ ((𝐴 ≼ (ℵ‘𝐵) ∧ (ℵ‘𝐵) ≺ (ℵ‘suc 𝐵)) → 𝐴 ≺ (ℵ‘suc 𝐵)) | |
3 | 2 | ex 416 | . . 3 ⊢ (𝐴 ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) ≺ (ℵ‘suc 𝐵) → 𝐴 ≺ (ℵ‘suc 𝐵))) |
4 | 1, 3 | syl5com 31 | . 2 ⊢ (𝐵 ∈ On → (𝐴 ≼ (ℵ‘𝐵) → 𝐴 ≺ (ℵ‘suc 𝐵))) |
5 | sdomdom 8520 | . . . . 5 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → 𝐴 ≼ (ℵ‘suc 𝐵)) | |
6 | alephon 9480 | . . . . . 6 ⊢ (ℵ‘suc 𝐵) ∈ On | |
7 | ondomen 9448 | . . . . . 6 ⊢ (((ℵ‘suc 𝐵) ∈ On ∧ 𝐴 ≼ (ℵ‘suc 𝐵)) → 𝐴 ∈ dom card) | |
8 | 6, 7 | mpan 689 | . . . . 5 ⊢ (𝐴 ≼ (ℵ‘suc 𝐵) → 𝐴 ∈ dom card) |
9 | cardid2 9366 | . . . . 5 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
10 | 5, 8, 9 | 3syl 18 | . . . 4 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → (card‘𝐴) ≈ 𝐴) |
11 | 10 | ensymd 8543 | . . 3 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → 𝐴 ≈ (card‘𝐴)) |
12 | alephnbtwn2 9483 | . . . . . 6 ⊢ ¬ ((ℵ‘𝐵) ≺ (card‘𝐴) ∧ (card‘𝐴) ≺ (ℵ‘suc 𝐵)) | |
13 | 12 | imnani 404 | . . . . 5 ⊢ ((ℵ‘𝐵) ≺ (card‘𝐴) → ¬ (card‘𝐴) ≺ (ℵ‘suc 𝐵)) |
14 | ensdomtr 8637 | . . . . . 6 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≺ (ℵ‘suc 𝐵)) → (card‘𝐴) ≺ (ℵ‘suc 𝐵)) | |
15 | 10, 14 | mpancom 687 | . . . . 5 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → (card‘𝐴) ≺ (ℵ‘suc 𝐵)) |
16 | 13, 15 | nsyl3 140 | . . . 4 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → ¬ (ℵ‘𝐵) ≺ (card‘𝐴)) |
17 | cardon 9357 | . . . . 5 ⊢ (card‘𝐴) ∈ On | |
18 | alephon 9480 | . . . . 5 ⊢ (ℵ‘𝐵) ∈ On | |
19 | domtriord 8647 | . . . . 5 ⊢ (((card‘𝐴) ∈ On ∧ (ℵ‘𝐵) ∈ On) → ((card‘𝐴) ≼ (ℵ‘𝐵) ↔ ¬ (ℵ‘𝐵) ≺ (card‘𝐴))) | |
20 | 17, 18, 19 | mp2an 691 | . . . 4 ⊢ ((card‘𝐴) ≼ (ℵ‘𝐵) ↔ ¬ (ℵ‘𝐵) ≺ (card‘𝐴)) |
21 | 16, 20 | sylibr 237 | . . 3 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → (card‘𝐴) ≼ (ℵ‘𝐵)) |
22 | endomtr 8550 | . . 3 ⊢ ((𝐴 ≈ (card‘𝐴) ∧ (card‘𝐴) ≼ (ℵ‘𝐵)) → 𝐴 ≼ (ℵ‘𝐵)) | |
23 | 11, 21, 22 | syl2anc 587 | . 2 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → 𝐴 ≼ (ℵ‘𝐵)) |
24 | 4, 23 | impbid1 228 | 1 ⊢ (𝐵 ∈ On → (𝐴 ≼ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘suc 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∈ wcel 2111 class class class wbr 5030 dom cdm 5519 Oncon0 6159 suc csuc 6161 ‘cfv 6324 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 cardccrd 9348 ℵcale 9349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-har 9005 df-card 9352 df-aleph 9353 |
This theorem is referenced by: alephsuc2 9491 alephreg 9993 |
Copyright terms: Public domain | W3C validator |