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Theorem oaordi 8585
Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
Assertion
Ref Expression
oaordi ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Proof of Theorem oaordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 6408 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21adantll 714 . . . 4 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → 𝐴 ∈ On)
3 eloni 6393 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
4 ordsucss 7839 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
53, 4syl 17 . . . . . . . 8 (𝐵 ∈ On → (𝐴𝐵 → suc 𝐴𝐵))
65ad2antlr 727 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴𝐵 → suc 𝐴𝐵))
7 onsucb 7838 . . . . . . . . . 10 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8 oveq2 7440 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
98sseq2d 4015 . . . . . . . . . . . . 13 (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
109imbi2d 340 . . . . . . . . . . . 12 (𝑥 = suc 𝐴 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴))))
11 oveq2 7440 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
1211sseq2d 4015 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))
1312imbi2d 340 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))))
14 oveq2 7440 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
1514sseq2d 4015 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
1615imbi2d 340 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
17 oveq2 7440 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
1817sseq2d 4015 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
1918imbi2d 340 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))
20 ssid 4005 . . . . . . . . . . . . 13 (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)
21202a1i 12 . . . . . . . . . . . 12 (suc 𝐴 ∈ On → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
22 sssucid 6463 . . . . . . . . . . . . . . . . 17 (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
23 sstr2 3989 . . . . . . . . . . . . . . . . 17 ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → ((𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
2422, 23mpi 20 . . . . . . . . . . . . . . . 16 ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦))
25 oasuc 8563 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2625ancoms 458 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2726sseq2d 4015 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
2824, 27imbitrrid 246 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
2928ex 412 . . . . . . . . . . . . . 14 (𝑦 ∈ On → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
3029ad2antrr 726 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑦) → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
3130a2d 29 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑦) → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
32 sucssel 6478 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
337, 32sylbir 235 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
34 limsuc 7871 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
3534biimpd 229 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 → suc 𝐴𝑥))
3633, 35sylan9r 508 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ suc 𝐴 ∈ On) → (suc 𝐴𝑥 → suc 𝐴𝑥))
3736imp 406 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → suc 𝐴𝑥)
38 oveq2 7440 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴))
3938ssiun2s 5047 . . . . . . . . . . . . . . . . 17 (suc 𝐴𝑥 → (𝐶 +o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 +o 𝑦))
4037, 39syl 17 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → (𝐶 +o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 +o 𝑦))
4140adantr 480 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 +o 𝑦))
42 vex 3483 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
43 oalim 8571 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4442, 43mpanr1 703 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4544ancoms 458 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥𝐶 ∈ On) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4645adantlr 715 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4746adantlr 715 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4841, 47sseqtrrd 4020 . . . . . . . . . . . . . 14 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥))
4948ex 412 . . . . . . . . . . . . 13 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)))
5049a1d 25 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → (∀𝑦𝑥 (suc 𝐴𝑦 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥))))
5110, 13, 16, 19, 21, 31, 50tfindsg 7883 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝐵) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
5251exp31 419 . . . . . . . . . 10 (𝐵 ∈ On → (suc 𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
537, 52biimtrid 242 . . . . . . . . 9 (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
5453com4r 94 . . . . . . . 8 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
5554imp31 417 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (suc 𝐴𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
56 oasuc 8563 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
5756sseq1d 4014 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58 ovex 7465 . . . . . . . . . 10 (𝐶 +o 𝐴) ∈ V
59 sucssel 6478 . . . . . . . . . 10 ((𝐶 +o 𝐴) ∈ V → (suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
6058, 59ax-mp 5 . . . . . . . . 9 (suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
6157, 60biimtrdi 253 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
6261adantlr 715 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
636, 55, 623syld 60 . . . . . 6 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
6463imp 406 . . . . 5 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
6564an32s 652 . . . 4 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
662, 65mpdan 687 . . 3 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
6766ex 412 . 2 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
6867ancoms 458 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  wss 3950   ciun 4990  Ord word 6382  Oncon0 6383  Lim wlim 6384  suc csuc 6385  (class class class)co 7432   +o coa 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-oadd 8511
This theorem is referenced by:  oaord  8586  oaass  8600  odi  8618  onexomgt  43258  onexoegt  43261  oaltublim  43308  oaordi3  43309  oacl2g  43348  tfsconcatfv2  43358  tfsconcatrn  43360  tfsconcatrev  43366  ofoafg  43372  oaun3lem1  43392  oaun3lem2  43393  oadif1  43398  naddwordnexlem0  43414  naddwordnexlem3  43417  naddwordnexlem4  43419  oaltom  43423
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