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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 6921 | . . 3 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 270 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 3291 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 6928 | . . . . . . . . 9 ⊢ 1o ∈ N | |
5 | ltpiord 6932 | . . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) | |
6 | 4, 5 | mpan2 417 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) |
7 | df-1o 6195 | . . . . . . . . . 10 ⊢ 1o = suc ∅ | |
8 | 7 | eleq2i 2155 | . . . . . . . . 9 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 4240 | . . . . . . . . 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 291 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
13 | 12 | ord 679 | . . . . 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 2298 | . 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 665 = wceq 1290 ∈ wcel 1439 ≠ wne 2256 ∅c0 3287 class class class wbr 3851 suc csuc 4201 ωcom 4418 1oc1o 6188 Ncnpi 6885 <N clti 6888 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-eprel 4125 df-suc 4207 df-iom 4419 df-xp 4457 df-1o 6195 df-ni 6917 df-lti 6920 |
This theorem is referenced by: caucvgsr 7401 |
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