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| Mirrors > Home > MPE Home > Th. List > oacl | Structured version Visualization version GIF version | ||
| Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. Remark 2.8 of [Schloeder] p. 5. (Contributed by NM, 5-May-1995.) |
| Ref | Expression |
|---|---|
| oacl | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq2 7439 | . . . 4 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅)) | |
| 2 | 1 | eleq1d 2826 | . . 3 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o ∅) ∈ On)) |
| 3 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦)) | |
| 4 | 3 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝑦) ∈ On)) |
| 5 | oveq2 7439 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦)) | |
| 6 | 5 | eleq1d 2826 | . . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o suc 𝑦) ∈ On)) |
| 7 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵)) | |
| 8 | 7 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝐵) ∈ On)) |
| 9 | oa0 8554 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) | |
| 10 | 9 | eleq1d 2826 | . . . 4 ⊢ (𝐴 ∈ On → ((𝐴 +o ∅) ∈ On ↔ 𝐴 ∈ On)) |
| 11 | 10 | ibir 268 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) ∈ On) |
| 12 | onsuc 7831 | . . . . 5 ⊢ ((𝐴 +o 𝑦) ∈ On → suc (𝐴 +o 𝑦) ∈ On) | |
| 13 | oasuc 8562 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦)) | |
| 14 | 13 | eleq1d 2826 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o suc 𝑦) ∈ On ↔ suc (𝐴 +o 𝑦) ∈ On)) |
| 15 | 12, 14 | imbitrrid 246 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On)) |
| 16 | 15 | expcom 413 | . . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On))) |
| 17 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 18 | iunon 8379 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On) | |
| 19 | 17, 18 | mpan 690 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On) |
| 20 | oalim 8570 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦)) | |
| 21 | 17, 20 | mpanr1 703 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦)) |
| 22 | 21 | eleq1d 2826 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On)) |
| 23 | 19, 22 | imbitrrid 246 | . . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On)) |
| 24 | 23 | expcom 413 | . . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On))) |
| 25 | 2, 4, 6, 8, 11, 16, 24 | tfinds3 7886 | . 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) ∈ On)) |
| 26 | 25 | impcom 407 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∅c0 4333 ∪ ciun 4991 Oncon0 6384 Lim wlim 6385 suc csuc 6386 (class class class)co 7431 +o coa 8503 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-oadd 8510 |
| This theorem is referenced by: omcl 8574 oaord 8585 oacan 8586 oaword 8587 oawordri 8588 oawordeulem 8592 oalimcl 8598 oaass 8599 oaf1o 8601 odi 8617 omopth2 8622 oeoalem 8634 oeoa 8635 oancom 9691 cantnfvalf 9705 dfac12lem2 10185 djunum 10236 wunex3 10781 rdgeqoa 37371 oaomoecl 43291 oawordex2 43339 omabs2 43345 tfsconcatlem 43349 tfsconcatun 43350 tfsconcatfv2 43353 tfsconcatfv 43354 tfsconcatrn 43355 tfsconcatb0 43357 tfsconcatrev 43361 ofoafg 43367 oaun3lem1 43387 oaun3lem2 43388 oaun3lem3 43389 oaun3lem4 43390 oadif1 43393 oaun2 43394 oaun3 43395 naddgeoa 43407 naddwordnexlem3 43412 oawordex3 43413 naddwordnexlem4 43414 oa1cl 43460 |
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