Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7514 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶))) | |
2 | 1 | 3comr 1265 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶))) |
3 | oaord 7514 | . . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ∈ 𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵))) | |
4 | 3 | 3com13 1262 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ∈ 𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵))) |
5 | 2, 4 | orbi12d 742 | . . 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 307 | . 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 5650 | . . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
8 | eloni 5650 | . . . 4 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
9 | ordtri3 5676 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) | |
10 | 7, 8, 9 | syl2an 493 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
11 | 10 | 3adant1 1072 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
12 | oacl 7502 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On) | |
13 | eloni 5650 | . . . . 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 7502 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +𝑜 𝐶) ∈ On) | |
16 | eloni 5650 | . . . . 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 5676 | . . . 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 493 | . . 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 1373 | . 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 300 | 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 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Ord word 5639 Oncon0 5640 (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: oawordeulem 7521 |
Copyright terms: Public domain | W3C validator |