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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 8466 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7420 | . . 3 ⊢ (𝐴 +o 1o) = (𝐴 +o suc ∅) |
3 | peano1 7879 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onasuc 8528 | . . . 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 2785 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc (𝐴 +o ∅)) |
7 | oa0 8516 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) | |
8 | suceq 6431 | . . 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 2773 | 1 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∅c0 4323 Oncon0 6365 suc csuc 6367 (class class class)co 7409 ωcom 7855 1oc1o 8459 +o coa 8463 |
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 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 |
This theorem is referenced by: o1p1e2 8540 o2p2e4 8541 om1r 8543 omlimcl 8578 oneo 8581 oeeui 8602 nnneo 8654 nneob 8655 oancom 9646 ttrcltr 9711 indpi 10902 oaabsb 42092 oa1un 42245 tr3dom 42327 sucomisnotcard 42343 nna1iscard 42344 |
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