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Mirrors > Home > MPE Home > Th. List > oldfi | Structured version Visualization version GIF version |
Description: The old set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.) |
Ref | Expression |
---|---|
oldfi | ⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7905 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | oldval 27902 | . . 3 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
4 | madef 27904 | . . . . 5 ⊢ M :On⟶𝒫 No | |
5 | ffun 6749 | . . . . 5 ⊢ ( M :On⟶𝒫 No → Fun M ) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ Fun M |
7 | nnfi 9229 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
8 | imafi 9377 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ Fin) → ( M “ 𝐴) ∈ Fin) | |
9 | 6, 7, 8 | sylancr 586 | . . 3 ⊢ (𝐴 ∈ ω → ( M “ 𝐴) ∈ Fin) |
10 | elnn 7910 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω) | |
11 | 10 | ancoms 458 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ω) |
12 | madefi 27959 | . . . . . 6 ⊢ (𝑥 ∈ ω → ( M ‘𝑥) ∈ Fin) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ 𝐴) → ( M ‘𝑥) ∈ Fin) |
14 | 13 | ralrimiva 3148 | . . . 4 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 ( M ‘𝑥) ∈ Fin) |
15 | onss 7816 | . . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
16 | 1, 15 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ On) |
17 | 4 | fdmi 6757 | . . . . . 6 ⊢ dom M = On |
18 | 16, 17 | sseqtrrdi 4054 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ dom M ) |
19 | funimass4 6985 | . . . . 5 ⊢ ((Fun M ∧ 𝐴 ⊆ dom M ) → (( M “ 𝐴) ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 ( M ‘𝑥) ∈ Fin)) | |
20 | 6, 18, 19 | sylancr 586 | . . . 4 ⊢ (𝐴 ∈ ω → (( M “ 𝐴) ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 ( M ‘𝑥) ∈ Fin)) |
21 | 14, 20 | mpbird 257 | . . 3 ⊢ (𝐴 ∈ ω → ( M “ 𝐴) ⊆ Fin) |
22 | unifi 9408 | . . 3 ⊢ ((( M “ 𝐴) ∈ Fin ∧ ( M “ 𝐴) ⊆ Fin) → ∪ ( M “ 𝐴) ∈ Fin) | |
23 | 9, 21, 22 | syl2anc 583 | . 2 ⊢ (𝐴 ∈ ω → ∪ ( M “ 𝐴) ∈ Fin) |
24 | 3, 23 | eqeltrd 2838 | 1 ⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∀wral 3063 ⊆ wss 3970 𝒫 cpw 4622 ∪ cuni 4931 dom cdm 5699 “ cima 5702 Oncon0 6394 Fun wfun 6566 ⟶wf 6568 ‘cfv 6572 ωcom 7899 Fincfn 8999 No csur 27693 M cmade 27890 O cold 27891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-ac2 10528 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-fin 9003 df-card 10004 df-acn 10007 df-ac 10181 df-no 27696 df-slt 27697 df-bday 27698 df-sslt 27835 df-scut 27837 df-made 27895 df-old 27896 |
This theorem is referenced by: (None) |
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