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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 4602. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nndceq | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6472 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
2 | elirr 4525 | . . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | eleq2 2234 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | mtbii 669 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
5 | 4 | con2i 622 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵) |
6 | 5 | olcd 729 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
7 | orc 707 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
8 | elirr 4525 | . . . . . . 7 ⊢ ¬ 𝐵 ∈ 𝐵 | |
9 | eleq2 2234 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵)) | |
10 | 8, 9 | mtbiri 670 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴) |
11 | 10 | con2i 622 | . . . . 5 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = 𝐵) |
12 | 11 | olcd 729 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
13 | 6, 7, 12 | 3jaoi 1298 | . . 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 830 | . 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 703 DECID wdc 829 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 ωcom 4574 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 |
This theorem is referenced by: nndifsnid 6486 fidceq 6847 unsnfidcex 6897 unsnfidcel 6898 nninfwlporlemd 7148 nninfwlporlem 7149 nninfwlpoimlemg 7151 nninfwlpoimlemginf 7152 enqdc 7323 nninfsellemdc 14043 |
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