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 6658 | . . 3 ⊢ (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅)) | |
2 | id 22 | . . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
3 | 1, 2 | eqeq12d 2637 | . 2 ⊢ (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅)) |
4 | oveq2 6658 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦)) | |
5 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2637 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦)) |
7 | oveq2 6658 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦)) | |
8 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
9 | 7, 8 | eqeq12d 2637 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦)) |
10 | oveq2 6658 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴)) | |
11 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2637 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴)) |
13 | 0elon 5778 | . . 3 ⊢ ∅ ∈ On | |
14 | oa0 7596 | . . 3 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (∅ +𝑜 ∅) = ∅ |
16 | oasuc 7604 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦)) | |
17 | 13, 16 | mpan 706 | . . . 4 ⊢ (𝑦 ∈ On → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦)) |
18 | suceq 5790 | . . . 4 ⊢ ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦) | |
19 | 17, 18 | sylan9eq 2676 | . . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +𝑜 𝑦) = 𝑦) → (∅ +𝑜 suc 𝑦) = suc 𝑦) |
20 | 19 | ex 450 | . 2 ⊢ (𝑦 ∈ On → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦)) |
21 | iuneq2 4537 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦) | |
22 | uniiun 4573 | . . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
23 | 21, 22 | syl6eqr 2674 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑥) |
24 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | oalim 7612 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦)) | |
26 | 13, 25 | mpan 706 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦)) |
27 | 24, 26 | mpan 706 | . . . 4 ⊢ (Lim 𝑥 → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦)) |
28 | limuni 5785 | . . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥) | |
29 | 27, 28 | eqeq12d 2637 | . . 3 ⊢ (Lim 𝑥 → ((∅ +𝑜 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑥)) |
30 | 23, 29 | syl5ibr 236 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 𝑥) = 𝑥)) |
31 | 3, 6, 9, 12, 15, 20, 30 | tfinds 7059 | 1 ⊢ (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∅c0 3915 ∪ cuni 4436 ∪ ciun 4520 Oncon0 5723 Lim wlim 5724 suc csuc 5725 (class class class)co 6650 +𝑜 coa 7557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 |
This theorem is referenced by: om1 7622 oaword2 7633 oeeui 7682 oaabs2 7725 cantnfp1 8578 |
Copyright terms: Public domain | W3C validator |