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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 8774 and alephsuc2 8786. Equality can be obtained by taking the card of the right-hand side then using alephcard 8776 and carden 9252. (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 8786 | . . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)}) | |
2 | alephcard 8776 | . . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴) | |
3 | alephon 8775 | . . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On | |
4 | onenon 8658 | . . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ dom card |
6 | cardval2 8700 | . . . . . . . 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 2634 | . . . . . 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 3691 | . . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})) |
11 | difrab 3860 | . . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} | |
12 | bren2 7872 | . . . . . . 7 ⊢ (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))) | |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))) |
14 | 13 | rabbiia 3161 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))} |
15 | 11, 14 | eqtr4i 2635 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} |
16 | 10, 15 | syl6req 2661 | . . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴))) |
17 | alephon 8775 | . . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On | |
18 | onenon 8658 | . . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card) | |
19 | 17, 18 | mp1i 13 | . . . 4 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card) |
20 | sucelon 6909 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | |
21 | alephgeom 8788 | . . . . . 6 ⊢ (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) | |
22 | 20, 21 | bitri 263 | . . . . 5 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴)) |
23 | fvex 6113 | . . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ V | |
24 | ssdomg 7887 | . . . . . 6 ⊢ ((ℵ‘suc 𝐴) ∈ V → (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴))) | |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴)) |
26 | 22, 25 | sylbi 206 | . . . 4 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴)) |
27 | alephordilem1 8779 | . . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) | |
28 | infdif 8914 | . . . 4 ⊢ (((ℵ‘suc 𝐴) ∈ dom card ∧ ω ≼ (ℵ‘suc 𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴)) | |
29 | 19, 26, 27, 28 | syl3anc 1318 | . . 3 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴)) |
30 | 16, 29 | eqbrtrd 4605 | . 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴)) |
31 | 30 | ensymd 7893 | 1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 Oncon0 5640 suc csuc 5642 ‘cfv 5804 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 cardccrd 8644 ℵcale 8645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-har 8346 df-card 8648 df-aleph 8649 df-cda 8873 |
This theorem is referenced by: (None) |
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