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Mirrors > Home > MPE Home > Th. List > nna0 | Structured version Visualization version 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 7238 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | oa0 7768 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2140 ∅c0 4059 Oncon0 5885 (class class class)co 6815 ωcom 7232 +𝑜 coa 7728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-oadd 7735 |
This theorem is referenced by: nnacl 7863 nnacom 7869 nnaass 7874 nndi 7875 nnmsucr 7877 nnaword1 7881 nnmordi 7883 nnawordex 7889 nnaordex 7890 ackbij1lem13 9267 addnidpi 9936 1lt2pi 9940 hashgadd 13379 |
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