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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 9211 and alephsuc2 9223. Equality can be obtained by taking the card of the right-hand side then using alephcard 9213 and carden 9695. (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 9223 | . . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)}) | |
2 | alephcard 9213 | . . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴) | |
3 | alephon 9212 | . . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On | |
4 | onenon 9095 | . . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ dom card |
6 | cardval2 9137 | . . . . . . . 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 2851 | . . . . . 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 3958 | . . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})) |
11 | difrab 4132 | . . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} | |
12 | bren2 8259 | . . . . . 6 ⊢ (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))) | |
13 | 12 | rabbii 3398 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} |
14 | 11, 13 | eqtr4i 2852 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} |
15 | 10, 14 | syl6req 2878 | . . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴))) |
16 | alephon 9212 | . . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On | |
17 | onenon 9095 | . . . . 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 7283 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | |
20 | alephgeom 9225 | . . . . . 6 ⊢ (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) | |
21 | 19, 20 | bitri 267 | . . . . 5 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) |
22 | fvex 6450 | . . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ V | |
23 | ssdomg 8274 | . . . . . 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 209 | . . . 4 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴)) |
26 | alephordilem1 9216 | . . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) | |
27 | infdif 9353 | . . . 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 1494 | . . 3 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴)) |
29 | 15, 28 | eqbrtrd 4897 | . 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴)) |
30 | 29 | ensymd 8279 | 1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {crab 3121 Vcvv 3414 ∖ cdif 3795 ⊆ wss 3798 class class class wbr 4875 dom cdm 5346 Oncon0 5967 suc csuc 5969 ‘cfv 6127 ωcom 7331 ≈ cen 8225 ≼ cdom 8226 ≺ csdm 8227 cardccrd 9081 ℵcale 9082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-oi 8691 df-har 8739 df-card 9085 df-aleph 9086 df-cda 9312 |
This theorem is referenced by: (None) |
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