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Mirrors > Home > ILE Home > Th. List > uzenom | GIF version |
Description: An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
uzinf.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
uzenom | ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
2 | eqid 2189 | . . . . 5 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) | |
3 | 1, 2 | frec2uzf1od 10432 | . . . 4 ⊢ (𝑀 ∈ ℤ → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀):ω–1-1-onto→(ℤ≥‘𝑀)) |
4 | omex 4607 | . . . . 5 ⊢ ω ∈ V | |
5 | 4 | f1oen 6780 | . . . 4 ⊢ (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀):ω–1-1-onto→(ℤ≥‘𝑀) → ω ≈ (ℤ≥‘𝑀)) |
6 | 3, 5 | syl 14 | . . 3 ⊢ (𝑀 ∈ ℤ → ω ≈ (ℤ≥‘𝑀)) |
7 | uzinf.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
8 | 6, 7 | breqtrrdi 4060 | . 2 ⊢ (𝑀 ∈ ℤ → ω ≈ 𝑍) |
9 | 8 | ensymd 6804 | 1 ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 ↦ cmpt 4079 ωcom 4604 –1-1-onto→wf1o 5231 ‘cfv 5232 (class class class)co 5892 freccfrec 6410 ≈ cen 6759 1c1 7837 + caddc 7839 ℤcz 9278 ℤ≥cuz 9553 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-addass 7938 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-ltadd 7952 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-recs 6325 df-frec 6411 df-er 6554 df-en 6762 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-inn 8945 df-n0 9202 df-z 9279 df-uz 9554 |
This theorem is referenced by: (None) |
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