![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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. (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 7447 | . . . 4 ⊢ 1𝑜 = suc ∅ | |
2 | 1 | oveq2i 6560 | . . 3 ⊢ (𝐴 +𝑜 1𝑜) = (𝐴 +𝑜 suc ∅) |
3 | peano1 6977 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onasuc 7495 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 +𝑜 suc ∅) = suc (𝐴 +𝑜 ∅)) | |
5 | 3, 4 | mpan2 703 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 suc ∅) = suc (𝐴 +𝑜 ∅)) |
6 | 2, 5 | syl5eq 2656 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc (𝐴 +𝑜 ∅)) |
7 | oa0 7483 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴) | |
8 | suceq 5707 | . . 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 2644 | 1 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∅c0 3874 Oncon0 5640 suc csuc 5642 (class class class)co 6549 ωcom 6957 1𝑜c1o 7440 +𝑜 coa 7444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 |
This theorem is referenced by: o1p1e2 7507 o2p2e4 7508 om1r 7510 omlimcl 7545 oneo 7548 oeeui 7569 nnneo 7618 nneob 7619 oancom 8431 indpi 9608 |
Copyright terms: Public domain | W3C validator |