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Mirrors > Home > ILE Home > Th. List > nndceq | GIF version |
Description: Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4635. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nndceq | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6519 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
2 | elirr 4558 | . . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | eleq2 2253 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | mtbii 675 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
5 | 4 | con2i 628 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵) |
6 | 5 | olcd 735 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
7 | orc 713 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
8 | elirr 4558 | . . . . . . 7 ⊢ ¬ 𝐵 ∈ 𝐵 | |
9 | eleq2 2253 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵)) | |
10 | 8, 9 | mtbiri 676 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴) |
11 | 10 | con2i 628 | . . . . 5 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = 𝐵) |
12 | 11 | olcd 735 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
13 | 6, 7, 12 | 3jaoi 1314 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
14 | 1, 13 | syl 14 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
15 | df-dc 836 | . 2 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
16 | 14, 15 | sylibr 134 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 DECID wdc 835 ∨ w3o 979 = wceq 1364 ∈ wcel 2160 ωcom 4607 |
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-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 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-uni 3825 df-int 3860 df-tr 4117 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 |
This theorem is referenced by: nndifsnid 6533 fidceq 6898 unsnfidcex 6949 unsnfidcel 6950 nninfwlporlemd 7201 nninfwlporlem 7202 nninfwlpoimlemg 7204 nninfwlpoimlemginf 7205 2onetap 7285 2omotaplemap 7287 enqdc 7391 xpscf 12826 nninfsellemdc 15238 |
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