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Mirrors > Home > MPE Home > Th. List > alephsuc3 | Structured version Visualization version GIF version |
Description: An alternate representation of a successor aleph. Compare alephsuc 9479 and alephsuc2 9491. Equality can be obtained by taking the card of the right-hand side then using alephcard 9481 and carden 9962. (Contributed by NM, 23-Oct-2004.) |
Ref | Expression |
---|---|
alephsuc3 | ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephsuc2 9491 | . . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)}) | |
2 | alephcard 9481 | . . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴) | |
3 | alephon 9480 | . . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On | |
4 | onenon 9362 | . . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ dom card |
6 | cardval2 9404 | . . . . . . . 8 ⊢ ((ℵ‘𝐴) ∈ dom card → (card‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)} |
8 | 2, 7 | eqtr3i 2823 | . . . . . 6 ⊢ (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)} |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) |
10 | 1, 9 | difeq12d 4051 | . . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})) |
11 | difrab 4229 | . . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} | |
12 | bren2 8523 | . . . . . 6 ⊢ (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))) | |
13 | 12 | rabbii 3420 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} |
14 | 11, 13 | eqtr4i 2824 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} |
15 | 10, 14 | eqtr2di 2850 | . . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴))) |
16 | alephon 9480 | . . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On | |
17 | onenon 9362 | . . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card) | |
18 | 16, 17 | mp1i 13 | . . . 4 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card) |
19 | sucelon 7512 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | |
20 | alephgeom 9493 | . . . . . 6 ⊢ (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) | |
21 | 19, 20 | bitri 278 | . . . . 5 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) |
22 | fvex 6658 | . . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ V | |
23 | ssdomg 8538 | . . . . . 6 ⊢ ((ℵ‘suc 𝐴) ∈ V → (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴))) | |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴)) |
25 | 21, 24 | sylbi 220 | . . . 4 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴)) |
26 | alephordilem1 9484 | . . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) | |
27 | infdif 9620 | . . . 4 ⊢ (((ℵ‘suc 𝐴) ∈ dom card ∧ ω ≼ (ℵ‘suc 𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴)) | |
28 | 18, 25, 26, 27 | syl3anc 1368 | . . 3 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴)) |
29 | 15, 28 | eqbrtrd 5052 | . 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴)) |
30 | 29 | ensymd 8543 | 1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 class class class wbr 5030 dom cdm 5519 Oncon0 6159 suc csuc 6161 ‘cfv 6324 ωcom 7560 ≈ 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-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-har 9005 df-dju 9314 df-card 9352 df-aleph 9353 |
This theorem is referenced by: (None) |
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