![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alephsuc2 | Structured version Visualization version GIF version |
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9536 function by transfinite recursion, starting from ω. Using this theorem we could define the aleph function with {𝑧 ∈ On ∣ 𝑧 ≼ 𝑥} in place of ∩ {𝑧 ∈ On ∣ 𝑥 ≺ 𝑧} in df-aleph 9932. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
Ref | Expression |
---|---|
alephsuc2 | ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephon 10061 | . . . . . . 7 ⊢ (ℵ‘suc 𝐴) ∈ On | |
2 | 1 | oneli 6469 | . . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ∈ On) |
3 | alephcard 10062 | . . . . . . . . 9 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) | |
4 | iscard 9967 | . . . . . . . . 9 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴))) | |
5 | 3, 4 | mpbi 229 | . . . . . . . 8 ⊢ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)) |
6 | 5 | simpri 485 | . . . . . . 7 ⊢ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴) |
7 | 6 | rspec 3239 | . . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ≺ (ℵ‘suc 𝐴)) |
8 | 2, 7 | jca 511 | . . . . 5 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴))) |
9 | sdomel 9121 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴))) | |
10 | 1, 9 | mpan2 688 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴))) |
11 | 10 | imp 406 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)) → 𝑦 ∈ (ℵ‘suc 𝐴)) |
12 | 8, 11 | impbii 208 | . . . 4 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴))) |
13 | breq1 5142 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ (ℵ‘suc 𝐴) ↔ 𝑦 ≺ (ℵ‘suc 𝐴))) | |
14 | 13 | elrab 3676 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴))) |
15 | 12, 14 | bitr4i 278 | . . 3 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)}) |
16 | 15 | eqriv 2721 | . 2 ⊢ (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)} |
17 | alephsucdom 10071 | . . 3 ⊢ (𝐴 ∈ On → (𝑥 ≼ (ℵ‘𝐴) ↔ 𝑥 ≺ (ℵ‘suc 𝐴))) | |
18 | 17 | rabbidv 3432 | . 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)}) |
19 | 16, 18 | eqtr4id 2783 | 1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 {crab 3424 class class class wbr 5139 Oncon0 6355 suc csuc 6357 ‘cfv 6534 ≼ cdom 8934 ≺ csdm 8935 cardccrd 9927 ℵcale 9928 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-oi 9502 df-har 9549 df-card 9931 df-aleph 9932 |
This theorem is referenced by: alephsuc3 10572 |
Copyright terms: Public domain | W3C validator |