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Mirrors > Home > ILE Home > Th. List > oa1suc | GIF version |
Description: Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
oa1suc | ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 6395 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 5864 | . . 3 ⊢ (𝐴 +o 1o) = (𝐴 +o suc ∅) |
3 | peano1 4578 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onasuc 6445 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 +o suc ∅) = suc (𝐴 +o ∅)) | |
5 | 3, 4 | mpan2 423 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +o suc ∅) = suc (𝐴 +o ∅)) |
6 | 2, 5 | eqtrid 2215 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc (𝐴 +o ∅)) |
7 | oa0 6436 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) | |
8 | suceq 4387 | . . 3 ⊢ ((𝐴 +o ∅) = 𝐴 → suc (𝐴 +o ∅) = suc 𝐴) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝐴 ∈ On → suc (𝐴 +o ∅) = suc 𝐴) |
10 | 6, 9 | eqtrd 2203 | 1 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ∅c0 3414 Oncon0 4348 suc csuc 4350 ωcom 4574 (class class class)co 5853 1oc1o 6388 +o coa 6392 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-oadd 6399 |
This theorem is referenced by: o1p1e2 6447 oawordriexmid 6449 nnaordex 6507 indpi 7304 prarloclemlo 7456 |
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