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Theorem oaord 8014
Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)
Assertion
Ref Expression
oaord ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Proof of Theorem oaord
StepHypRef Expression
1 oaordi 8013 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
213adant1 1121 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
3 oveq2 7015 . . . . . 6 (𝐴 = 𝐵 → (𝐶 +o 𝐴) = (𝐶 +o 𝐵))
43a1i 11 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 = 𝐵 → (𝐶 +o 𝐴) = (𝐶 +o 𝐵)))
5 oaordi 8013 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐴 → (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴)))
653adant2 1122 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐴 → (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴)))
74, 6orim12d 957 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 +o 𝐴) = (𝐶 +o 𝐵) ∨ (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴))))
87con3d 155 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ((𝐶 +o 𝐴) = (𝐶 +o 𝐵) ∨ (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 df-3an 1080 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On))
10 ancom 461 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ↔ (𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)))
11 anandi 672 . . . . . 6 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ↔ ((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)))
129, 10, 113bitri 298 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ↔ ((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)))
13 oacl 8002 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ On)
14 eloni 6068 . . . . . . 7 ((𝐶 +o 𝐴) ∈ On → Ord (𝐶 +o 𝐴))
1513, 14syl 17 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → Ord (𝐶 +o 𝐴))
16 oacl 8002 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 +o 𝐵) ∈ On)
17 eloni 6068 . . . . . . 7 ((𝐶 +o 𝐵) ∈ On → Ord (𝐶 +o 𝐵))
1816, 17syl 17 . . . . . 6 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐶 +o 𝐵))
1915, 18anim12i 612 . . . . 5 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)) → (Ord (𝐶 +o 𝐴) ∧ Ord (𝐶 +o 𝐵)))
2012, 19sylbi 218 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (Ord (𝐶 +o 𝐴) ∧ Ord (𝐶 +o 𝐵)))
21 ordtri2 6093 . . . 4 ((Ord (𝐶 +o 𝐴) ∧ Ord (𝐶 +o 𝐵)) → ((𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵) ↔ ¬ ((𝐶 +o 𝐴) = (𝐶 +o 𝐵) ∨ (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴))))
2220, 21syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵) ↔ ¬ ((𝐶 +o 𝐴) = (𝐶 +o 𝐵) ∨ (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴))))
23 eloni 6068 . . . . . 6 (𝐴 ∈ On → Ord 𝐴)
24 eloni 6068 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
2523, 24anim12i 612 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Ord 𝐴 ∧ Ord 𝐵))
26253adant3 1123 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (Ord 𝐴 ∧ Ord 𝐵))
27 ordtri2 6093 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2826, 27syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
298, 22, 283imtr4d 295 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵) → 𝐴𝐵))
302, 29impbid 213 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842  w3a 1078   = wceq 1520  wcel 2079  Ord word 6057  Oncon0 6058  (class class class)co 7007   +o coa 7941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-oadd 7948
This theorem is referenced by:  oacan  8015  oaword  8016  oaord1  8018  oa00  8026  oalimcl  8027  oaass  8028  odi  8046  oneo  8048  omeulem1  8049  omeulem2  8050  oeeui  8069  omxpenlem  8455  cantnflt  8970  cantnflem1d  8986  cantnflem1  8987
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