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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 521 | . . . 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 4527 | . . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑢 ∈ ω) | |
5 | 1, 3, 4 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑢 ∈ ω) |
6 | simprr 522 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ 𝐴) | |
7 | elnn 4527 | . . . 4 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑣 ∈ ω) | |
8 | 6, 3, 7 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ ω) |
9 | nntri3 6401 | . . 3 ⊢ ((𝑢 ∈ ω ∧ 𝑣 ∈ ω) → (𝑢 = 𝑣 ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢))) | |
10 | 5, 8, 9 | syl2anc 409 | . 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢))) |
11 | epel 4222 | . . . 4 ⊢ (𝑢 E 𝑣 ↔ 𝑢 ∈ 𝑣) | |
12 | 11 | notbii 658 | . . 3 ⊢ (¬ 𝑢 E 𝑣 ↔ ¬ 𝑢 ∈ 𝑣) |
13 | epel 4222 | . . . 4 ⊢ (𝑣 E 𝑢 ↔ 𝑣 ∈ 𝑢) | |
14 | 13 | notbii 658 | . . 3 ⊢ (¬ 𝑣 E 𝑢 ↔ ¬ 𝑣 ∈ 𝑢) |
15 | 12, 14 | anbi12i 456 | . 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 1481 class class class wbr 3937 E cep 4217 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-3or 964 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 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-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-tr 4035 df-eprel 4219 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 |
This theorem is referenced by: (None) |
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