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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 8156 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶))) | |
2 | 1 | 3comr 1122 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶))) |
3 | oaord 8156 | . . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ∈ 𝐵 ↔ (𝐴 +o 𝐶) ∈ (𝐴 +o 𝐵))) | |
4 | 3 | 3com13 1121 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ∈ 𝐵 ↔ (𝐴 +o 𝐶) ∈ (𝐴 +o 𝐵))) |
5 | 2, 4 | orbi12d 916 | . . 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 321 | . 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 6169 | . . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
8 | eloni 6169 | . . . 4 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
9 | ordtri3 6195 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) | |
10 | 7, 8, 9 | syl2an 598 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
11 | 10 | 3adant1 1127 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
12 | oacl 8143 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On) | |
13 | eloni 6169 | . . . . 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 8143 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +o 𝐶) ∈ On) | |
16 | eloni 6169 | . . . . 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 6195 | . . . 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 598 | . . 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 1347 | . 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 315 | 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 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Ord word 6158 Oncon0 6159 (class class class)co 7135 +o coa 8082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 |
This theorem is referenced by: oawordeulem 8163 |
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