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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 10098 | . . 3 ⊢ (𝐵 ∈ On → (ℵ‘𝐵) ≺ (ℵ‘suc 𝐵)) | |
2 | domsdomtr 9137 | . . . 4 ⊢ ((𝐴 ≼ (ℵ‘𝐵) ∧ (ℵ‘𝐵) ≺ (ℵ‘suc 𝐵)) → 𝐴 ≺ (ℵ‘suc 𝐵)) | |
3 | 2 | ex 411 | . . 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 9001 | . . . . 5 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → 𝐴 ≼ (ℵ‘suc 𝐵)) | |
6 | alephon 10094 | . . . . . 6 ⊢ (ℵ‘suc 𝐵) ∈ On | |
7 | ondomen 10062 | . . . . . 6 ⊢ (((ℵ‘suc 𝐵) ∈ On ∧ 𝐴 ≼ (ℵ‘suc 𝐵)) → 𝐴 ∈ dom card) | |
8 | 6, 7 | mpan 688 | . . . . 5 ⊢ (𝐴 ≼ (ℵ‘suc 𝐵) → 𝐴 ∈ dom card) |
9 | cardid2 9978 | . . . . 5 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
10 | 5, 8, 9 | 3syl 18 | . . . 4 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → (card‘𝐴) ≈ 𝐴) |
11 | 10 | ensymd 9026 | . . 3 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → 𝐴 ≈ (card‘𝐴)) |
12 | alephnbtwn2 10097 | . . . . . 6 ⊢ ¬ ((ℵ‘𝐵) ≺ (card‘𝐴) ∧ (card‘𝐴) ≺ (ℵ‘suc 𝐵)) | |
13 | 12 | imnani 399 | . . . . 5 ⊢ ((ℵ‘𝐵) ≺ (card‘𝐴) → ¬ (card‘𝐴) ≺ (ℵ‘suc 𝐵)) |
14 | ensdomtr 9138 | . . . . . 6 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≺ (ℵ‘suc 𝐵)) → (card‘𝐴) ≺ (ℵ‘suc 𝐵)) | |
15 | 10, 14 | mpancom 686 | . . . . 5 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → (card‘𝐴) ≺ (ℵ‘suc 𝐵)) |
16 | 13, 15 | nsyl3 138 | . . . 4 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → ¬ (ℵ‘𝐵) ≺ (card‘𝐴)) |
17 | cardon 9969 | . . . . 5 ⊢ (card‘𝐴) ∈ On | |
18 | alephon 10094 | . . . . 5 ⊢ (ℵ‘𝐵) ∈ On | |
19 | domtriord 9148 | . . . . 5 ⊢ (((card‘𝐴) ∈ On ∧ (ℵ‘𝐵) ∈ On) → ((card‘𝐴) ≼ (ℵ‘𝐵) ↔ ¬ (ℵ‘𝐵) ≺ (card‘𝐴))) | |
20 | 17, 18, 19 | mp2an 690 | . . . 4 ⊢ ((card‘𝐴) ≼ (ℵ‘𝐵) ↔ ¬ (ℵ‘𝐵) ≺ (card‘𝐴)) |
21 | 16, 20 | sylibr 233 | . . 3 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → (card‘𝐴) ≼ (ℵ‘𝐵)) |
22 | endomtr 9033 | . . 3 ⊢ ((𝐴 ≈ (card‘𝐴) ∧ (card‘𝐴) ≼ (ℵ‘𝐵)) → 𝐴 ≼ (ℵ‘𝐵)) | |
23 | 11, 21, 22 | syl2anc 582 | . 2 ⊢ (𝐴 ≺ (ℵ‘suc 𝐵) → 𝐴 ≼ (ℵ‘𝐵)) |
24 | 4, 23 | impbid1 224 | 1 ⊢ (𝐵 ∈ On → (𝐴 ≼ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘suc 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2098 class class class wbr 5149 dom cdm 5678 Oncon0 6371 suc csuc 6373 ‘cfv 6549 ≈ cen 8961 ≼ cdom 8962 ≺ csdm 8963 cardccrd 9960 ℵcale 9961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9535 df-har 9582 df-card 9964 df-aleph 9965 |
This theorem is referenced by: alephsuc2 10105 alephreg 10607 |
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