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Mirrors > Home > MPE Home > Th. List > oacan | Structured version Visualization version GIF version |
Description: Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.) |
Ref | Expression |
---|---|
oacan | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oaord 7627 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶))) | |
2 | 1 | 3comr 1273 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶))) |
3 | oaord 7627 | . . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ∈ 𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵))) | |
4 | 3 | 3com13 1270 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ∈ 𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵))) |
5 | 2, 4 | orbi12d 746 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵) ↔ ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶) ∨ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵)))) |
6 | 5 | notbid 308 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶) ∨ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵)))) |
7 | eloni 5733 | . . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
8 | eloni 5733 | . . . 4 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
9 | ordtri3 5759 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) | |
10 | 7, 8, 9 | syl2an 494 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
11 | 10 | 3adant1 1079 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
12 | oacl 7615 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On) | |
13 | eloni 5733 | . . . . 5 ⊢ ((𝐴 +𝑜 𝐵) ∈ On → Ord (𝐴 +𝑜 𝐵)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +𝑜 𝐵)) |
15 | oacl 7615 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +𝑜 𝐶) ∈ On) | |
16 | eloni 5733 | . . . . 5 ⊢ ((𝐴 +𝑜 𝐶) ∈ On → Ord (𝐴 +𝑜 𝐶)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 +𝑜 𝐶)) |
18 | ordtri3 5759 | . . . 4 ⊢ ((Ord (𝐴 +𝑜 𝐵) ∧ Ord (𝐴 +𝑜 𝐶)) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶) ∨ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵)))) | |
19 | 14, 17, 18 | syl2an 494 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶) ∨ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵)))) |
20 | 19 | 3impdi 1381 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶) ∨ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵)))) |
21 | 6, 11, 20 | 3bitr4rd 301 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Ord word 5722 Oncon0 5723 (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: oawordeulem 7634 |
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