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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. Lemma 2.14 of [Schloeder] p. 5. (Contributed by NM, 5-May-1995.) |
Ref | Expression |
---|---|
oa0r | ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7423 | . . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅)) | |
2 | id 22 | . . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
3 | 1, 2 | eqeq12d 2744 | . 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅)) |
4 | oveq2 7423 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦)) | |
5 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2744 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦)) |
7 | oveq2 7423 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦)) | |
8 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
9 | 7, 8 | eqeq12d 2744 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦)) |
10 | oveq2 7423 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴)) | |
11 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2744 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴)) |
13 | 0elon 6418 | . . 3 ⊢ ∅ ∈ On | |
14 | oa0 8531 | . . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (∅ +o ∅) = ∅ |
16 | oasuc 8539 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦)) | |
17 | 13, 16 | mpan 689 | . . . 4 ⊢ (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦)) |
18 | suceq 6430 | . . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦) | |
19 | 17, 18 | sylan9eq 2788 | . . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦) |
20 | 19 | ex 412 | . 2 ⊢ (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦)) |
21 | iuneq2 5011 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦) | |
22 | uniiun 5056 | . . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
23 | 21, 22 | eqtr4di 2786 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥) |
24 | vex 3474 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | oalim 8547 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦)) | |
26 | 13, 25 | mpan 689 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦)) |
27 | 24, 26 | mpan 689 | . . . 4 ⊢ (Lim 𝑥 → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦)) |
28 | limuni 6425 | . . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥) | |
29 | 27, 28 | eqeq12d 2744 | . . 3 ⊢ (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥)) |
30 | 23, 29 | imbitrrid 245 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥)) |
31 | 3, 6, 9, 12, 15, 20, 30 | tfinds 7859 | 1 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 ∅c0 4319 ∪ cuni 4904 ∪ ciun 4992 Oncon0 6364 Lim wlim 6365 suc csuc 6366 (class class class)co 7415 +o coa 8478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-oadd 8485 |
This theorem is referenced by: om1 8557 oaword2 8568 oeeui 8617 oaabs2 8664 cantnfp1 9699 oaordnrex 42715 oacl2g 42750 tfsconcat0i 42765 ofoaf 42775 ofoaid2 42779 |
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