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Mirrors > Home > MPE Home > Th. List > oa0 | Structured version Visualization version GIF version |
Description: Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. Definition 2.3 of [Schloeder] p. 4. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oa0 | ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6416 | . . 3 ⊢ ∅ ∈ On | |
2 | oav 8508 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 +o ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) |
4 | rdg0g 8424 | . 2 ⊢ (𝐴 ∈ On → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | eqtrd 2773 | 1 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4322 ↦ cmpt 5231 Oncon0 6362 suc csuc 6364 ‘cfv 6541 (class class class)co 7406 reccrdg 8406 +o coa 8460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-oadd 8467 |
This theorem is referenced by: oa1suc 8528 oacl 8532 oa0r 8535 om0r 8536 oawordri 8547 oaord1 8548 oaword1 8549 oawordeulem 8551 oa00 8556 oaass 8558 oarec 8559 odi 8576 oeoalem 8593 nna0 8601 nna0r 8606 nnm0r 8607 nnawordi 8618 cantnflt 9664 rdgeqoa 36240 oa0suclim 42011 cantnfresb 42060 dflim5 42065 omabs2 42068 tfsconcatb0 42080 ofoafo 42092 ofoaid1 42094 naddcnff 42098 naddcnffo 42100 oaun3lem1 42110 naddgeoa 42131 naddonnn 42132 naddwordnexlem4 42138 |
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