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Mirrors > Home > ILE Home > Th. List > Mathboxes > nnti | GIF version |
Description: Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.) |
Ref | Expression |
---|---|
nnti.a | ⊢ (𝜑 → 𝐴 ∈ ω) |
Ref | Expression |
---|---|
nnti | ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 520 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑢 ∈ 𝐴) | |
2 | nnti.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ω) | |
3 | 2 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝐴 ∈ ω) |
4 | elnn 4519 | . . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑢 ∈ ω) | |
5 | 1, 3, 4 | syl2anc 408 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑢 ∈ ω) |
6 | simprr 521 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ 𝐴) | |
7 | elnn 4519 | . . . 4 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑣 ∈ ω) | |
8 | 6, 3, 7 | syl2anc 408 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ ω) |
9 | nntri3 6393 | . . 3 ⊢ ((𝑢 ∈ ω ∧ 𝑣 ∈ ω) → (𝑢 = 𝑣 ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢))) | |
10 | 5, 8, 9 | syl2anc 408 | . 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢))) |
11 | epel 4214 | . . . 4 ⊢ (𝑢 E 𝑣 ↔ 𝑢 ∈ 𝑣) | |
12 | 11 | notbii 657 | . . 3 ⊢ (¬ 𝑢 E 𝑣 ↔ ¬ 𝑢 ∈ 𝑣) |
13 | epel 4214 | . . . 4 ⊢ (𝑣 E 𝑢 ↔ 𝑣 ∈ 𝑢) | |
14 | 13 | notbii 657 | . . 3 ⊢ (¬ 𝑣 E 𝑢 ↔ ¬ 𝑣 ∈ 𝑢) |
15 | 12, 14 | anbi12i 455 | . 2 ⊢ ((¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢) ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢)) |
16 | 10, 15 | syl6bbr 197 | 1 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 class class class wbr 3929 E cep 4209 ωcom 4504 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-tr 4027 df-eprel 4211 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 |
This theorem is referenced by: (None) |
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