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Mirrors > Home > ILE Home > Th. List > ennnfonelemen | GIF version |
Description: Lemma for ennnfone 11938. The result. (Contributed by Jim Kingdon, 16-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemh.dceq | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
ennnfonelemh.f | ⊢ (𝜑 → 𝐹:ω–onto→𝐴) |
ennnfonelemh.ne | ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) |
ennnfonelemh.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {〈dom 𝑥, (𝐹‘𝑦)〉}))) |
ennnfonelemh.n | ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
ennnfonelemh.j | ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) |
ennnfonelemh.h | ⊢ 𝐻 = seq0(𝐺, 𝐽) |
ennnfone.l | ⊢ 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) |
Ref | Expression |
---|---|
ennnfonelemen | ⊢ (𝜑 → 𝐴 ≈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemh.dceq | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) | |
2 | ennnfonelemh.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
3 | ennnfonelemh.ne | . . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) | |
4 | ennnfonelemh.g | . . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {〈dom 𝑥, (𝐹‘𝑦)〉}))) | |
5 | ennnfonelemh.n | . . . . . . 7 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
6 | ennnfonelemh.j | . . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
7 | ennnfonelemh.h | . . . . . . 7 ⊢ 𝐻 = seq0(𝐺, 𝐽) | |
8 | ennnfone.l | . . . . . . 7 ⊢ 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ennnfonelemf1 11931 | . . . . . 6 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | ennnfonelemdm 11933 | . . . . . . 7 ⊢ (𝜑 → dom 𝐿 = ω) |
11 | f1eq2 5324 | . . . . . . 7 ⊢ (dom 𝐿 = ω → (𝐿:dom 𝐿–1-1→𝐴 ↔ 𝐿:ω–1-1→𝐴)) | |
12 | 10, 11 | syl 14 | . . . . . 6 ⊢ (𝜑 → (𝐿:dom 𝐿–1-1→𝐴 ↔ 𝐿:ω–1-1→𝐴)) |
13 | 9, 12 | mpbid 146 | . . . . 5 ⊢ (𝜑 → 𝐿:ω–1-1→𝐴) |
14 | 1, 2, 3, 4, 5, 6, 7, 8 | ennnfonelemrn 11932 | . . . . 5 ⊢ (𝜑 → ran 𝐿 = 𝐴) |
15 | dff1o5 5376 | . . . . 5 ⊢ (𝐿:ω–1-1-onto→𝐴 ↔ (𝐿:ω–1-1→𝐴 ∧ ran 𝐿 = 𝐴)) | |
16 | 13, 14, 15 | sylanbrc 413 | . . . 4 ⊢ (𝜑 → 𝐿:ω–1-1-onto→𝐴) |
17 | omex 4507 | . . . . 5 ⊢ ω ∈ V | |
18 | 17 | f1oen 6653 | . . . 4 ⊢ (𝐿:ω–1-1-onto→𝐴 → ω ≈ 𝐴) |
19 | 16, 18 | syl 14 | . . 3 ⊢ (𝜑 → ω ≈ 𝐴) |
20 | 19 | ensymd 6677 | . 2 ⊢ (𝜑 → 𝐴 ≈ ω) |
21 | nnenom 10207 | . . 3 ⊢ ℕ ≈ ω | |
22 | 21 | ensymi 6676 | . 2 ⊢ ω ≈ ℕ |
23 | entr 6678 | . 2 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ) | |
24 | 20, 22, 23 | sylancl 409 | 1 ⊢ (𝜑 → 𝐴 ≈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∀wral 2416 ∃wrex 2417 ∪ cun 3069 ∅c0 3363 ifcif 3474 {csn 3527 〈cop 3530 ∪ ciun 3813 class class class wbr 3929 ↦ cmpt 3989 suc csuc 4287 ωcom 4504 ◡ccnv 4538 dom cdm 4539 ran crn 4540 “ cima 4542 –1-1→wf1 5120 –onto→wfo 5121 –1-1-onto→wf1o 5122 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 freccfrec 6287 ↑pm cpm 6543 ≈ cen 6632 0cc0 7620 1c1 7621 + caddc 7623 − cmin 7933 ℕcn 8720 ℕ0cn0 8977 ℤcz 9054 seqcseq 10218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-er 6429 df-pm 6545 df-en 6635 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 |
This theorem is referenced by: ennnfonelemnn0 11935 |
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