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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 7865 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | oa0 8522 | . 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 1540 ∈ wcel 2105 ∅c0 4322 Oncon0 6364 (class class class)co 7412 ωcom 7859 +o coa 8469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-oadd 8476 |
This theorem is referenced by: nnacl 8617 nnacom 8623 nnaass 8628 nndi 8629 nnmsucr 8631 nnaword1 8635 nnmordi 8637 nnawordex 8643 nnaordex 8644 nnadju 10198 ackbij1lem13 10233 addnidpi 10902 1lt2pi 10906 hashgadd 14344 precsexlem6 27912 precsexlem7 27913 naddcnfid1 42432 |
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