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Mirrors > Home > ILE Home > Th. List > nna0 | GIF version |
Description: Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
Ref | Expression |
---|---|
nna0 | ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 4523 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | oa0 6353 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∅c0 3363 Oncon0 4285 ωcom 4504 (class class class)co 5774 +o coa 6310 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-irdg 6267 df-oadd 6317 |
This theorem is referenced by: nnacl 6376 nnacom 6380 nnaass 6381 nndi 6382 nnmsucr 6384 nnaordi 6404 nnmordi 6412 nnaordex 6423 nnawordex 6424 addnidpig 7144 1lt2pi 7148 archnqq 7225 prarloclemarch2 7227 nq0a0 7265 prarloclem3 7305 omgadd 10548 hashunlem 10550 |
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