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Mirrors > Home > MPE Home > Th. List > alephiso | Structured version Visualization version GIF version |
Description: Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.) |
Ref | Expression |
---|---|
alephiso | ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephfnon 9568 | . . . . . 6 ⊢ ℵ Fn On | |
2 | isinfcard 9595 | . . . . . . . 8 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ 𝑥 ∈ ran ℵ) | |
3 | 2 | bicomi 227 | . . . . . . 7 ⊢ (𝑥 ∈ ran ℵ ↔ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)) |
4 | 3 | abbi2i 2872 | . . . . . 6 ⊢ ran ℵ = {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} |
5 | df-fo 6346 | . . . . . 6 ⊢ (ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ↔ (ℵ Fn On ∧ ran ℵ = {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})) | |
6 | 1, 4, 5 | mpbir2an 711 | . . . . 5 ⊢ ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} |
7 | fof 6593 | . . . . 5 ⊢ (ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} → ℵ:On⟶{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ ℵ:On⟶{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} |
9 | aleph11 9587 | . . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) ↔ 𝑦 = 𝑧)) | |
10 | 9 | biimpd 232 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧)) |
11 | 10 | rgen2 3116 | . . . 4 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧) |
12 | dff13 7027 | . . . 4 ⊢ (ℵ:On–1-1→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ↔ (ℵ:On⟶{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ∧ ∀𝑦 ∈ On ∀𝑧 ∈ On ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧))) | |
13 | 8, 11, 12 | mpbir2an 711 | . . 3 ⊢ ℵ:On–1-1→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} |
14 | df-f1o 6347 | . . 3 ⊢ (ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ↔ (ℵ:On–1-1→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ∧ ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})) | |
15 | 13, 6, 14 | mpbir2an 711 | . 2 ⊢ ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} |
16 | alephord2 9579 | . . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧))) | |
17 | epel 5438 | . . . 4 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
18 | fvex 6690 | . . . . 5 ⊢ (ℵ‘𝑧) ∈ V | |
19 | 18 | epeli 5437 | . . . 4 ⊢ ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)) |
20 | 16, 17, 19 | 3bitr4g 317 | . . 3 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧))) |
21 | 20 | rgen2 3116 | . 2 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)) |
22 | df-isom 6349 | . 2 ⊢ (ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}) ↔ (ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ∧ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)))) | |
23 | 15, 21, 22 | mpbir2an 711 | 1 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {cab 2717 ∀wral 3054 ⊆ wss 3844 class class class wbr 5031 E cep 5434 ran crn 5527 Oncon0 6173 Fn wfn 6335 ⟶wf 6336 –1-1→wf1 6337 –onto→wfo 6338 –1-1-onto→wf1o 6339 ‘cfv 6340 Isom wiso 6341 ωcom 7602 cardccrd 9440 ℵcale 9441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-om 7603 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-oi 9050 df-har 9097 df-card 9444 df-aleph 9445 |
This theorem is referenced by: alephiso2 40733 |
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