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Mirrors > Home > ILE Home > Th. List > unennn | GIF version |
Description: The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.) |
Ref | Expression |
---|---|
unennn | ⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddennn 12446 | . . . . . 6 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ≈ ℕ | |
2 | 1 | ensymi 6809 | . . . . 5 ⊢ ℕ ≈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} |
3 | entr 6811 | . . . . 5 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) → 𝐴 ≈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) | |
4 | 2, 3 | mpan2 425 | . . . 4 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) |
5 | 4 | 3ad2ant1 1020 | . . 3 ⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ≈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) |
6 | evenennn 12447 | . . . . . 6 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ | |
7 | 6 | ensymi 6809 | . . . . 5 ⊢ ℕ ≈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} |
8 | entr 6811 | . . . . 5 ⊢ ((𝐵 ≈ ℕ ∧ ℕ ≈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) → 𝐵 ≈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) | |
9 | 7, 8 | mpan2 425 | . . . 4 ⊢ (𝐵 ≈ ℕ → 𝐵 ≈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) |
10 | 9 | 3ad2ant2 1021 | . . 3 ⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ≈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) |
11 | simp3 1001 | . . 3 ⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅) | |
12 | inrab 3422 | . . . . 5 ⊢ ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∩ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) = {𝑧 ∈ ℕ ∣ (¬ 2 ∥ 𝑧 ∧ 2 ∥ 𝑧)} | |
13 | pm3.24 694 | . . . . . . . 8 ⊢ ¬ (2 ∥ 𝑧 ∧ ¬ 2 ∥ 𝑧) | |
14 | ancom 266 | . . . . . . . 8 ⊢ ((2 ∥ 𝑧 ∧ ¬ 2 ∥ 𝑧) ↔ (¬ 2 ∥ 𝑧 ∧ 2 ∥ 𝑧)) | |
15 | 13, 14 | mtbi 671 | . . . . . . 7 ⊢ ¬ (¬ 2 ∥ 𝑧 ∧ 2 ∥ 𝑧) |
16 | 15 | rgenw 2545 | . . . . . 6 ⊢ ∀𝑧 ∈ ℕ ¬ (¬ 2 ∥ 𝑧 ∧ 2 ∥ 𝑧) |
17 | rabeq0 3467 | . . . . . 6 ⊢ ({𝑧 ∈ ℕ ∣ (¬ 2 ∥ 𝑧 ∧ 2 ∥ 𝑧)} = ∅ ↔ ∀𝑧 ∈ ℕ ¬ (¬ 2 ∥ 𝑧 ∧ 2 ∥ 𝑧)) | |
18 | 16, 17 | mpbir 146 | . . . . 5 ⊢ {𝑧 ∈ ℕ ∣ (¬ 2 ∥ 𝑧 ∧ 2 ∥ 𝑧)} = ∅ |
19 | 12, 18 | eqtri 2210 | . . . 4 ⊢ ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∩ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) = ∅ |
20 | 19 | a1i 9 | . . 3 ⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∩ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) = ∅) |
21 | unen 6843 | . . 3 ⊢ (((𝐴 ≈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝐵 ≈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∩ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) = ∅)) → (𝐴 ∪ 𝐵) ≈ ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∪ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧})) | |
22 | 5, 10, 11, 20, 21 | syl22anc 1250 | . 2 ⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∪ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧})) |
23 | unrab 3421 | . . 3 ⊢ ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∪ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) = {𝑧 ∈ ℕ ∣ (¬ 2 ∥ 𝑧 ∨ 2 ∥ 𝑧)} | |
24 | rabid2 2667 | . . . 4 ⊢ (ℕ = {𝑧 ∈ ℕ ∣ (¬ 2 ∥ 𝑧 ∨ 2 ∥ 𝑧)} ↔ ∀𝑧 ∈ ℕ (¬ 2 ∥ 𝑧 ∨ 2 ∥ 𝑧)) | |
25 | nnz 9303 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ) | |
26 | 2z 9312 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
27 | zdvdsdc 11854 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑧 ∈ ℤ) → DECID 2 ∥ 𝑧) | |
28 | 26, 27 | mpan 424 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → DECID 2 ∥ 𝑧) |
29 | exmiddc 837 | . . . . . 6 ⊢ (DECID 2 ∥ 𝑧 → (2 ∥ 𝑧 ∨ ¬ 2 ∥ 𝑧)) | |
30 | 25, 28, 29 | 3syl 17 | . . . . 5 ⊢ (𝑧 ∈ ℕ → (2 ∥ 𝑧 ∨ ¬ 2 ∥ 𝑧)) |
31 | 30 | orcomd 730 | . . . 4 ⊢ (𝑧 ∈ ℕ → (¬ 2 ∥ 𝑧 ∨ 2 ∥ 𝑧)) |
32 | 24, 31 | mprgbir 2548 | . . 3 ⊢ ℕ = {𝑧 ∈ ℕ ∣ (¬ 2 ∥ 𝑧 ∨ 2 ∥ 𝑧)} |
33 | 23, 32 | eqtr4i 2213 | . 2 ⊢ ({𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∪ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) = ℕ |
34 | 22, 33 | breqtrdi 4059 | 1 ⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∀wral 2468 {crab 2472 ∪ cun 3142 ∩ cin 3143 ∅c0 3437 class class class wbr 4018 ≈ cen 6765 ℕcn 8950 2c2 9001 ℤcz 9284 ∥ cdvds 11829 |
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-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 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-rmo 2476 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-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-er 6560 df-en 6768 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-n0 9208 df-z 9285 df-q 9652 df-rp 9686 df-fl 10303 df-mod 10356 df-dvds 11830 |
This theorem is referenced by: znnen 12452 |
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