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Mirrors > Home > ILE Home > Th. List > ennnfonelemen | GIF version |
Description: Lemma for ennnfone 12420. 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 12413 | . . . . . 6 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | ennnfonelemdm 12415 | . . . . . . 7 ⊢ (𝜑 → dom 𝐿 = ω) |
11 | f1eq2 5417 | . . . . . . 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 147 | . . . . 5 ⊢ (𝜑 → 𝐿:ω–1-1→𝐴) |
14 | 1, 2, 3, 4, 5, 6, 7, 8 | ennnfonelemrn 12414 | . . . . 5 ⊢ (𝜑 → ran 𝐿 = 𝐴) |
15 | dff1o5 5470 | . . . . 5 ⊢ (𝐿:ω–1-1-onto→𝐴 ↔ (𝐿:ω–1-1→𝐴 ∧ ran 𝐿 = 𝐴)) | |
16 | 13, 14, 15 | sylanbrc 417 | . . . 4 ⊢ (𝜑 → 𝐿:ω–1-1-onto→𝐴) |
17 | omex 4592 | . . . . 5 ⊢ ω ∈ V | |
18 | 17 | f1oen 6758 | . . . 4 ⊢ (𝐿:ω–1-1-onto→𝐴 → ω ≈ 𝐴) |
19 | 16, 18 | syl 14 | . . 3 ⊢ (𝜑 → ω ≈ 𝐴) |
20 | 19 | ensymd 6782 | . 2 ⊢ (𝜑 → 𝐴 ≈ ω) |
21 | nnenom 10431 | . . 3 ⊢ ℕ ≈ ω | |
22 | 21 | ensymi 6781 | . 2 ⊢ ω ≈ ℕ |
23 | entr 6783 | . 2 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ) | |
24 | 20, 22, 23 | sylancl 413 | 1 ⊢ (𝜑 → 𝐴 ≈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ∃wrex 2456 ∪ cun 3127 ∅c0 3422 ifcif 3534 {csn 3592 〈cop 3595 ∪ ciun 3886 class class class wbr 4003 ↦ cmpt 4064 suc csuc 4365 ωcom 4589 ◡ccnv 4625 dom cdm 4626 ran crn 4627 “ cima 4629 –1-1→wf1 5213 –onto→wfo 5214 –1-1-onto→wf1o 5215 ‘cfv 5216 (class class class)co 5874 ∈ cmpo 5876 freccfrec 6390 ↑pm cpm 6648 ≈ cen 6737 0cc0 7810 1c1 7811 + caddc 7813 − cmin 8126 ℕcn 8917 ℕ0cn0 9174 ℤcz 9251 seqcseq 10442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-er 6534 df-pm 6650 df-en 6740 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 df-uz 9527 df-seqfrec 10443 |
This theorem is referenced by: ennnfonelemnn0 12417 |
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