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Mirrors > Home > ILE Home > Th. List > ennnfonelemen | GIF version |
Description: Lemma for ennnfone 12301. 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 12294 | . . . . . 6 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | ennnfonelemdm 12296 | . . . . . . 7 ⊢ (𝜑 → dom 𝐿 = ω) |
11 | f1eq2 5383 | . . . . . . 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 12295 | . . . . 5 ⊢ (𝜑 → ran 𝐿 = 𝐴) |
15 | dff1o5 5435 | . . . . 5 ⊢ (𝐿:ω–1-1-onto→𝐴 ↔ (𝐿:ω–1-1→𝐴 ∧ ran 𝐿 = 𝐴)) | |
16 | 13, 14, 15 | sylanbrc 414 | . . . 4 ⊢ (𝜑 → 𝐿:ω–1-1-onto→𝐴) |
17 | omex 4564 | . . . . 5 ⊢ ω ∈ V | |
18 | 17 | f1oen 6716 | . . . 4 ⊢ (𝐿:ω–1-1-onto→𝐴 → ω ≈ 𝐴) |
19 | 16, 18 | syl 14 | . . 3 ⊢ (𝜑 → ω ≈ 𝐴) |
20 | 19 | ensymd 6740 | . 2 ⊢ (𝜑 → 𝐴 ≈ ω) |
21 | nnenom 10359 | . . 3 ⊢ ℕ ≈ ω | |
22 | 21 | ensymi 6739 | . 2 ⊢ ω ≈ ℕ |
23 | entr 6741 | . 2 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ) | |
24 | 20, 22, 23 | sylancl 410 | 1 ⊢ (𝜑 → 𝐴 ≈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 DECID wdc 824 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ∀wral 2442 ∃wrex 2443 ∪ cun 3109 ∅c0 3404 ifcif 3515 {csn 3570 〈cop 3573 ∪ ciun 3860 class class class wbr 3976 ↦ cmpt 4037 suc csuc 4337 ωcom 4561 ◡ccnv 4597 dom cdm 4598 ran crn 4599 “ cima 4601 –1-1→wf1 5179 –onto→wfo 5180 –1-1-onto→wf1o 5181 ‘cfv 5182 (class class class)co 5836 ∈ cmpo 5838 freccfrec 6349 ↑pm cpm 6606 ≈ cen 6695 0cc0 7744 1c1 7745 + caddc 7747 − cmin 8060 ℕcn 8848 ℕ0cn0 9105 ℤcz 9182 seqcseq 10370 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-er 6492 df-pm 6608 df-en 6698 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-seqfrec 10371 |
This theorem is referenced by: ennnfonelemnn0 12298 |
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