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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 6557 | . . 3 ⊢ (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅)) | |
2 | id 22 | . . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
3 | 1, 2 | eqeq12d 2625 | . 2 ⊢ (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅)) |
4 | oveq2 6557 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦)) | |
5 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2625 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦)) |
7 | oveq2 6557 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦)) | |
8 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
9 | 7, 8 | eqeq12d 2625 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦)) |
10 | oveq2 6557 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴)) | |
11 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2625 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴)) |
13 | 0elon 5695 | . . 3 ⊢ ∅ ∈ On | |
14 | oa0 7483 | . . 3 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (∅ +𝑜 ∅) = ∅ |
16 | oasuc 7491 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦)) | |
17 | 13, 16 | mpan 702 | . . . 4 ⊢ (𝑦 ∈ On → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦)) |
18 | suceq 5707 | . . . 4 ⊢ ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦) | |
19 | 17, 18 | sylan9eq 2664 | . . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +𝑜 𝑦) = 𝑦) → (∅ +𝑜 suc 𝑦) = suc 𝑦) |
20 | 19 | ex 449 | . 2 ⊢ (𝑦 ∈ On → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦)) |
21 | iuneq2 4473 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦) | |
22 | uniiun 4509 | . . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
23 | 21, 22 | syl6eqr 2662 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑥) |
24 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | oalim 7499 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦)) | |
26 | 13, 25 | mpan 702 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦)) |
27 | 24, 26 | mpan 702 | . . . 4 ⊢ (Lim 𝑥 → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦)) |
28 | limuni 5702 | . . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥) | |
29 | 27, 28 | eqeq12d 2625 | . . 3 ⊢ (Lim 𝑥 → ((∅ +𝑜 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑥)) |
30 | 23, 29 | syl5ibr 235 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 𝑥) = 𝑥)) |
31 | 3, 6, 9, 12, 15, 20, 30 | tfinds 6951 | 1 ⊢ (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∅c0 3874 ∪ cuni 4372 ∪ ciun 4455 Oncon0 5640 Lim wlim 5641 suc csuc 5642 (class class class)co 6549 +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-rep 4699 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-oadd 7451 |
This theorem is referenced by: om1 7509 oaword2 7520 oeeui 7569 oaabs2 7612 cantnfp1 8461 |
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