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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 4539. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nndceq | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6397 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
2 | elirr 4464 | . . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | eleq2 2204 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | mtbii 664 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
5 | 4 | con2i 617 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵) |
6 | 5 | olcd 724 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
7 | orc 702 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
8 | elirr 4464 | . . . . . . 7 ⊢ ¬ 𝐵 ∈ 𝐵 | |
9 | eleq2 2204 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵)) | |
10 | 8, 9 | mtbiri 665 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴) |
11 | 10 | con2i 617 | . . . . 5 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = 𝐵) |
12 | 11 | olcd 724 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
13 | 6, 7, 12 | 3jaoi 1282 | . . 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 821 | . 2 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
16 | 14, 15 | sylibr 133 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 820 ∨ w3o 962 = wceq 1332 ∈ wcel 1481 ωcom 4512 |
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 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 |
This theorem is referenced by: nndifsnid 6411 fidceq 6771 unsnfidcex 6816 unsnfidcel 6817 enqdc 7193 nninfsellemdc 13381 |
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