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Mirrors > Home > MPE Home > Th. List > oa1suc | Structured version Visualization version GIF version |
Description: Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. Remark 2.4 of [Schloeder] p. 4. (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 8417 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7373 | . . 3 ⊢ (𝐴 +o 1o) = (𝐴 +o suc ∅) |
3 | peano1 7830 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onasuc 8479 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 +o suc ∅) = suc (𝐴 +o ∅)) | |
5 | 3, 4 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +o suc ∅) = suc (𝐴 +o ∅)) |
6 | 2, 5 | eqtrid 2789 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc (𝐴 +o ∅)) |
7 | oa0 8467 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) | |
8 | suceq 6388 | . . 3 ⊢ ((𝐴 +o ∅) = 𝐴 → suc (𝐴 +o ∅) = suc 𝐴) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝐴 ∈ On → suc (𝐴 +o ∅) = suc 𝐴) |
10 | 6, 9 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∅c0 4287 Oncon0 6322 suc csuc 6324 (class class class)co 7362 ωcom 7807 1oc1o 8410 +o coa 8414 |
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 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
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 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 |
This theorem is referenced by: o1p1e2 8491 o2p2e4 8492 o2p2e4OLD 8493 om1r 8495 omlimcl 8530 oneo 8533 oeeui 8554 nnneo 8606 nneob 8607 oancom 9594 ttrcltr 9659 indpi 10850 oaabsb 41658 oa1un 41792 tr3dom 41874 sucomisnotcard 41890 nna1iscard 41891 |
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