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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 526 | . . . 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 4590 | . . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑢 ∈ ω) | |
5 | 1, 3, 4 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑢 ∈ ω) |
6 | simprr 527 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ 𝐴) | |
7 | elnn 4590 | . . . 4 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑣 ∈ ω) | |
8 | 6, 3, 7 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → 𝑣 ∈ ω) |
9 | nntri3 6476 | . . 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 4277 | . . . 4 ⊢ (𝑢 E 𝑣 ↔ 𝑢 ∈ 𝑣) | |
12 | 11 | notbii 663 | . . 3 ⊢ (¬ 𝑢 E 𝑣 ↔ ¬ 𝑢 ∈ 𝑣) |
13 | epel 4277 | . . . 4 ⊢ (𝑣 E 𝑢 ↔ 𝑣 ∈ 𝑢) | |
14 | 13 | notbii 663 | . . 3 ⊢ (¬ 𝑣 E 𝑢 ↔ ¬ 𝑣 ∈ 𝑢) |
15 | 12, 14 | anbi12i 457 | . 2 ⊢ ((¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢) ↔ (¬ 𝑢 ∈ 𝑣 ∧ ¬ 𝑣 ∈ 𝑢)) |
16 | 10, 15 | bitr4di 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 2141 class class class wbr 3989 E cep 4272 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-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 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-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-eprel 4274 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 |
This theorem is referenced by: (None) |
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