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Mirrors > Home > ILE Home > Th. List > omiunct | GIF version |
Description: The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12120 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.) |
Ref | Expression |
---|---|
omiunct.cc | ⊢ (𝜑 → CCHOICE) |
omiunct.g | ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) |
Ref | Expression |
---|---|
omiunct | ⊢ (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omiunct.cc | . . . 4 ⊢ (𝜑 → CCHOICE) | |
2 | omiunct.g | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) | |
3 | 1, 2 | omctfn 12123 | . . 3 ⊢ (𝜑 → ∃𝑓(𝑓 Fn ω ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o))) |
4 | exsimpr 1595 | . . 3 ⊢ (∃𝑓(𝑓 Fn ω ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) → ∃𝑓∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) |
6 | omct 7047 | . . 3 ⊢ ∃𝑘 𝑘:ω–onto→(ω ⊔ 1o) | |
7 | simpr 109 | . . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) ∧ 𝑘:ω–onto→(ω ⊔ 1o)) → 𝑘:ω–onto→(ω ⊔ 1o)) | |
8 | simplr 520 | . . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) ∧ 𝑘:ω–onto→(ω ⊔ 1o)) → ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) | |
9 | 7, 8 | ctiunctal 12121 | . . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) ∧ 𝑘:ω–onto→(ω ⊔ 1o)) → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o)) |
10 | 9 | expcom 115 | . . . 4 ⊢ (𝑘:ω–onto→(ω ⊔ 1o) → ((𝜑 ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o))) |
11 | 10 | exlimiv 1575 | . . 3 ⊢ (∃𝑘 𝑘:ω–onto→(ω ⊔ 1o) → ((𝜑 ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o))) |
12 | 6, 11 | ax-mp 5 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)) → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o)) |
13 | 5, 12 | exlimddv 1875 | 1 ⊢ (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1469 ∈ wcel 2125 ∀wral 2432 ∪ ciun 3845 ωcom 4543 Fn wfn 5158 –onto→wfo 5161 ‘cfv 5163 1oc1o 6346 ⊔ cdju 6967 CCHOICEwacc 7161 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-xor 1355 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-1st 6078 df-2nd 6079 df-recs 6242 df-frec 6328 df-1o 6353 df-er 6469 df-map 6584 df-en 6675 df-dju 6968 df-inl 6977 df-inr 6978 df-case 7014 df-cc 7162 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-fz 9891 df-fl 10147 df-mod 10200 df-seqfrec 10323 df-exp 10397 df-dvds 11661 |
This theorem is referenced by: (None) |
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