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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 6372 | . . 3 ⊢ ∅ ∈ On | |
2 | oav 8458 | . . 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 8374 | . 2 ⊢ (𝐴 ∈ On → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∅c0 4283 ↦ cmpt 5189 Oncon0 6318 suc csuc 6320 ‘cfv 6497 (class class class)co 7358 reccrdg 8356 +o coa 8410 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-oadd 8417 |
This theorem is referenced by: oa1suc 8478 oacl 8482 oa0r 8485 om0r 8486 oawordri 8498 oaord1 8499 oaword1 8500 oawordeulem 8502 oa00 8507 oaass 8509 oarec 8510 odi 8527 oeoalem 8544 nna0 8552 nna0r 8557 nnm0r 8558 nnawordi 8569 cantnflt 9609 rdgeqoa 35844 oa0suclim 41613 cantnfresb 41661 dflim5 41666 omabs2 41668 ofoafo 41673 ofoaid1 41675 naddcnff 41679 naddcnffo 41681 |
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