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Mirrors > Home > MPE Home > Th. List > Mathboxes > aleph1min | Structured version Visualization version GIF version |
Description: (ℵ‘1o) is the least uncountable ordinal. (Contributed by RP, 18-Nov-2023.) |
Ref | Expression |
---|---|
aleph1min | ⊢ (ℵ‘1o) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8102 | . . 3 ⊢ 1o = suc ∅ | |
2 | 1 | fveq2i 6673 | . 2 ⊢ (ℵ‘1o) = (ℵ‘suc ∅) |
3 | 0elon 6244 | . . . . 5 ⊢ ∅ ∈ On | |
4 | alephsuc 9494 | . . . . 5 ⊢ (∅ ∈ On → (ℵ‘suc ∅) = (har‘(ℵ‘∅))) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (ℵ‘suc ∅) = (har‘(ℵ‘∅)) |
6 | aleph0 9492 | . . . . 5 ⊢ (ℵ‘∅) = ω | |
7 | 6 | fveq2i 6673 | . . . 4 ⊢ (har‘(ℵ‘∅)) = (har‘ω) |
8 | 5, 7 | eqtri 2844 | . . 3 ⊢ (ℵ‘suc ∅) = (har‘ω) |
9 | omelon 9109 | . . . . 5 ⊢ ω ∈ On | |
10 | onenon 9378 | . . . . 5 ⊢ (ω ∈ On → ω ∈ dom card) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ω ∈ dom card |
12 | harval2 9426 | . . . 4 ⊢ (ω ∈ dom card → (har‘ω) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥}) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ (har‘ω) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥} |
14 | 8, 13 | eqtri 2844 | . 2 ⊢ (ℵ‘suc ∅) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥} |
15 | 2, 14 | eqtri 2844 | 1 ⊢ (ℵ‘1o) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {crab 3142 ∅c0 4291 ∩ cint 4876 class class class wbr 5066 dom cdm 5555 Oncon0 6191 suc csuc 6193 ‘cfv 6355 ωcom 7580 1oc1o 8095 ≺ csdm 8508 harchar 9020 cardccrd 9364 ℵcale 9365 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-oi 8974 df-har 9022 df-card 9368 df-aleph 9369 |
This theorem is referenced by: (None) |
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