![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > oa0r | Structured version Visualization version GIF version |
Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) |
Ref | Expression |
---|---|
oa0r | ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6984 | . . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅)) | |
2 | id 22 | . . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
3 | 1, 2 | eqeq12d 2793 | . 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅)) |
4 | oveq2 6984 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦)) | |
5 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2793 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦)) |
7 | oveq2 6984 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦)) | |
8 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
9 | 7, 8 | eqeq12d 2793 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦)) |
10 | oveq2 6984 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴)) | |
11 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2793 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴)) |
13 | 0elon 6082 | . . 3 ⊢ ∅ ∈ On | |
14 | oa0 7943 | . . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (∅ +o ∅) = ∅ |
16 | oasuc 7951 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦)) | |
17 | 13, 16 | mpan 677 | . . . 4 ⊢ (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦)) |
18 | suceq 6094 | . . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦) | |
19 | 17, 18 | sylan9eq 2834 | . . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦) |
20 | 19 | ex 405 | . 2 ⊢ (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦)) |
21 | iuneq2 4810 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦) | |
22 | uniiun 4848 | . . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
23 | 21, 22 | syl6eqr 2832 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥) |
24 | vex 3418 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | oalim 7959 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦)) | |
26 | 13, 25 | mpan 677 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦)) |
27 | 24, 26 | mpan 677 | . . . 4 ⊢ (Lim 𝑥 → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦)) |
28 | limuni 6089 | . . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥) | |
29 | 27, 28 | eqeq12d 2793 | . . 3 ⊢ (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥)) |
30 | 23, 29 | syl5ibr 238 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥)) |
31 | 3, 6, 9, 12, 15, 20, 30 | tfinds 7390 | 1 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3088 Vcvv 3415 ∅c0 4178 ∪ cuni 4712 ∪ ciun 4792 Oncon0 6029 Lim wlim 6030 suc csuc 6031 (class class class)co 6976 +o coa 7902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-oadd 7909 |
This theorem is referenced by: om1 7969 oaword2 7980 oeeui 8029 oaabs2 8072 cantnfp1 8938 |
Copyright terms: Public domain | W3C validator |