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Mirrors > Home > ILE Home > Th. List > nlt1pig | GIF version |
Description: No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nlt1pig | ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elni 7140 | . . 3 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 273 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 3372 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 7147 | . . . . . . . . 9 ⊢ 1o ∈ N | |
5 | ltpiord 7151 | . . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) | |
6 | 4, 5 | mpan2 422 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) |
7 | df-1o 6321 | . . . . . . . . . 10 ⊢ 1o = suc ∅ | |
8 | 7 | eleq2i 2207 | . . . . . . . . 9 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 4334 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) | |
10 | 8, 9 | syl5bb 191 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 ∈ 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
11 | 6, 10 | bitrd 187 | . . . . . . 7 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
12 | 11 | biimpa 294 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
13 | 12 | ord 714 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅)) |
14 | 3, 13 | mpi 15 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → 𝐴 = ∅) |
15 | 14 | ex 114 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 <N 1o → 𝐴 = ∅)) |
16 | 15 | necon3ad 2351 | . 2 ⊢ (𝐴 ∈ N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1o)) |
17 | 2, 16 | mpd 13 | 1 ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ∅c0 3368 class class class wbr 3937 suc csuc 4295 ωcom 4512 1oc1o 6314 Ncnpi 7104 <N clti 7107 |
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 |
This theorem depends on definitions: df-bi 116 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-eprel 4219 df-suc 4301 df-iom 4513 df-xp 4553 df-1o 6321 df-ni 7136 df-lti 7139 |
This theorem is referenced by: caucvgsr 7634 |
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