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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 2824 | . . 3 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o ∅) ∈ On)) |
3 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦)) | |
4 | 3 | eleq1d 2824 | . . 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 2824 | . . 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 2824 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝐵) ∈ On)) |
9 | oa0 8553 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) | |
10 | 9 | eleq1d 2824 | . . . 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 8561 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦)) | |
14 | 13 | eleq1d 2824 | . . . . 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 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
18 | iunon 8378 | . . . . . 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 8569 | . . . . . . 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 2824 | . . . . 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 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∅c0 4339 ∪ ciun 4996 Oncon0 6386 Lim wlim 6387 suc csuc 6388 (class class class)co 7431 +o coa 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 |
This theorem is referenced by: omcl 8573 oaord 8584 oacan 8585 oaword 8586 oawordri 8587 oawordeulem 8591 oalimcl 8597 oaass 8598 oaf1o 8600 odi 8616 omopth2 8621 oeoalem 8633 oeoa 8634 oancom 9689 cantnfvalf 9703 dfac12lem2 10183 djunum 10234 wunex3 10779 rdgeqoa 37353 oaomoecl 43268 oawordex2 43316 omabs2 43322 tfsconcatlem 43326 tfsconcatun 43327 tfsconcatfv2 43330 tfsconcatfv 43331 tfsconcatrn 43332 tfsconcatb0 43334 tfsconcatrev 43338 ofoafg 43344 oaun3lem1 43364 oaun3lem2 43365 oaun3lem3 43366 oaun3lem4 43367 oadif1 43370 oaun2 43371 oaun3 43372 naddgeoa 43384 naddwordnexlem3 43389 oawordex3 43390 naddwordnexlem4 43391 oa1cl 43437 |
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