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Mathbox for Jim Kingdon |
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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 529 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑢 ∈ 𝐴) | |
2 | nnti.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ω) | |
3 | 2 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝐴 ∈ ω) |
4 | elnn 4607 | . . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑢 ∈ ω) | |
5 | 1, 3, 4 | syl2anc 411 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑢 ∈ ω) |
6 | simprr 531 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ 𝐴) | |
7 | elnn 4607 | . . . 4 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑣 ∈ ω) | |
8 | 6, 3, 7 | syl2anc 411 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ ω) |
9 | nntri3 6500 | . . 3 ⊢ ((𝑢 ∈ ω ∧ 𝑣 ∈ ω) → (𝑢 = 𝑣 ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢))) | |
10 | 5, 8, 9 | syl2anc 411 | . 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢))) |
11 | epel 4294 | . . . 4 ⊢ (𝑢 E 𝑣 ↔ 𝑢 ∈ 𝑣) | |
12 | 11 | notbii 668 | . . 3 ⊢ (¬ 𝑢 E 𝑣 ↔ ¬ 𝑢 ∈ 𝑣) |
13 | epel 4294 | . . . 4 ⊢ (𝑣 E 𝑢 ↔ 𝑣 ∈ 𝑢) | |
14 | 13 | notbii 668 | . . 3 ⊢ (¬ 𝑣 E 𝑢 ↔ ¬ 𝑣 ∈ 𝑢) |
15 | 12, 14 | anbi12i 460 | . 2 ⊢ ((¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢) ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢)) |
16 | 10, 15 | bitr4di 198 | 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 104 ↔ wb 105 ∈ wcel 2148 class class class wbr 4005 E cep 4289 ωcom 4591 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-tr 4104 df-eprel 4291 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 |
This theorem is referenced by: (None) |
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