![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4459. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nndceq | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6294 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
2 | elirr 4385 | . . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | eleq2 2158 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | mtbii 637 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
5 | 4 | con2i 595 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵) |
6 | 5 | olcd 691 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
7 | orc 671 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
8 | elirr 4385 | . . . . . . 7 ⊢ ¬ 𝐵 ∈ 𝐵 | |
9 | eleq2 2158 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵)) | |
10 | 8, 9 | mtbiri 638 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴) |
11 | 10 | con2i 595 | . . . . 5 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = 𝐵) |
12 | 11 | olcd 691 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
13 | 6, 7, 12 | 3jaoi 1246 | . . 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 784 | . 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 667 DECID wdc 783 ∨ w3o 926 = wceq 1296 ∈ wcel 1445 ωcom 4433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-uni 3676 df-int 3711 df-tr 3959 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 |
This theorem is referenced by: nndifsnid 6306 fidceq 6665 unsnfidcex 6710 unsnfidcel 6711 enqdc 7017 nninfsellemdc 12616 |
Copyright terms: Public domain | W3C validator |