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Mirrors > Home > ILE Home > Th. List > nnppipi | GIF version |
Description: A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Ref | Expression |
---|---|
nnppipi | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → (𝐴 +o 𝐵) ∈ N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7271 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
2 | nnacl 6459 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω) | |
3 | 1, 2 | sylan2 284 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → (𝐴 +o 𝐵) ∈ ω) |
4 | nnaword2 6493 | . . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +o 𝐵)) | |
5 | 1, 4 | sylan 281 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +o 𝐵)) |
6 | 5 | ancoms 266 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → 𝐵 ⊆ (𝐴 +o 𝐵)) |
7 | elni2 7276 | . . . . 5 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
8 | 7 | simprbi 273 | . . . 4 ⊢ (𝐵 ∈ N → ∅ ∈ 𝐵) |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ∅ ∈ 𝐵) |
10 | 6, 9 | sseldd 3148 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ∅ ∈ (𝐴 +o 𝐵)) |
11 | elni2 7276 | . 2 ⊢ ((𝐴 +o 𝐵) ∈ N ↔ ((𝐴 +o 𝐵) ∈ ω ∧ ∅ ∈ (𝐴 +o 𝐵))) | |
12 | 3, 10, 11 | sylanbrc 415 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → (𝐴 +o 𝐵) ∈ N) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 ⊆ wss 3121 ∅c0 3414 ωcom 4574 (class class class)co 5853 +o coa 6392 Ncnpi 7234 |
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-coll 4104 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-dc 830 df-3an 975 df-tru 1351 df-fal 1354 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-ni 7266 |
This theorem is referenced by: nqpnq0nq 7415 prarloclemlt 7455 prarloclemlo 7456 prarloclemcalc 7464 |
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