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Theorem oacan 7782
Description: Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
Assertion
Ref Expression
oacan ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem oacan
StepHypRef Expression
1 oaord 7781 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐵𝐶 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶)))
213comr 1119 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶)))
3 oaord 7781 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵)))
433com13 1118 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵)))
52, 4orbi12d 904 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶𝐶𝐵) ↔ ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶) ∨ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵))))
65notbid 307 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵𝐶𝐶𝐵) ↔ ¬ ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 𝐶) ∨ (𝐴 +𝑜 𝐶) ∈ (𝐴 +𝑜 𝐵))))
7 eloni 5876 . . . 4 (𝐵 ∈ On → Ord 𝐵)
8 eloni 5876 . . . 4 (𝐶 ∈ On → Ord 𝐶)
9 ordtri3 5902 . . . 4 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
107, 8, 9syl2an 583 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
11103adant1 1124 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
12 oacl 7769 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
13 eloni 5876 . . . . 5 ((𝐴 +𝑜 𝐵) ∈ On → Ord (𝐴 +𝑜 𝐵))
1412, 13syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +𝑜 𝐵))
15 oacl 7769 . . . . 5 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +𝑜 𝐶) ∈ On)
16 eloni 5876 . . . . 5 ((𝐴 +𝑜 𝐶) ∈ On → Ord (𝐴 +𝑜 𝐶))
1715, 16syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 +𝑜 𝐶))
18 ordtri3 5902 . . . 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))))
1914, 17, 18syl2an 583 . . 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))))
20193impdi 1443 . 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))))
216, 11, 203bitr4rd 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  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  Ord word 5865  Oncon0 5866  (class class class)co 6793   +𝑜 coa 7710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717
This theorem is referenced by:  oawordeulem  7788
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