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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 9496 and alephsuc2 9508. Equality can be obtained by taking the card of the right-hand side then using alephcard 9498 and carden 9975. (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 9508 | . . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)}) | |
2 | alephcard 9498 | . . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴) | |
3 | alephon 9497 | . . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On | |
4 | onenon 9380 | . . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ dom card |
6 | cardval2 9422 | . . . . . . . 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 2848 | . . . . . 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 4102 | . . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})) |
11 | difrab 4279 | . . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} | |
12 | bren2 8542 | . . . . . 6 ⊢ (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))) | |
13 | 12 | rabbii 3475 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} |
14 | 11, 13 | eqtr4i 2849 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} |
15 | 10, 14 | syl6req 2875 | . . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴))) |
16 | alephon 9497 | . . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On | |
17 | onenon 9380 | . . . . 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 7534 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | |
20 | alephgeom 9510 | . . . . . 6 ⊢ (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) | |
21 | 19, 20 | bitri 277 | . . . . 5 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) |
22 | fvex 6685 | . . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ V | |
23 | ssdomg 8557 | . . . . . 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 219 | . . . 4 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴)) |
26 | alephordilem1 9501 | . . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) | |
27 | infdif 9633 | . . . 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 1367 | . . 3 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴)) |
29 | 15, 28 | eqbrtrd 5090 | . 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴)) |
30 | 29 | ensymd 8562 | 1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 class class class wbr 5068 dom cdm 5557 Oncon0 6193 suc csuc 6195 ‘cfv 6357 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 ≺ csdm 8510 cardccrd 9366 ℵcale 9367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-har 9024 df-dju 9332 df-card 9370 df-aleph 9371 |
This theorem is referenced by: (None) |
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