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Theorem oaordi 8545
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 6389 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21adantll 712 . . . 4 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → 𝐴 ∈ On)
3 eloni 6374 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
4 ordsucss 7805 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
53, 4syl 17 . . . . . . . 8 (𝐵 ∈ On → (𝐴𝐵 → suc 𝐴𝐵))
65ad2antlr 725 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴𝐵 → suc 𝐴𝐵))
7 onsucb 7804 . . . . . . . . . 10 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8 oveq2 7416 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
98sseq2d 4014 . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
1211sseq2d 4014 . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
1514sseq2d 4014 . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
1817sseq2d 4014 . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . 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 6444 . . . . . . . . . . . . . . . . 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 8523 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2625ancoms 459 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2726sseq2d 4014 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
2824, 27imbitrrid 245 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
2928ex 413 . . . . . . . . . . . . . 14 (𝑦 ∈ On → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
3029ad2antrr 724 . . . . . . . . . . . . 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 6459 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
337, 32sylbir 234 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
34 limsuc 7837 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
3534biimpd 228 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 → suc 𝐴𝑥))
3633, 35sylan9r 509 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ suc 𝐴 ∈ On) → (suc 𝐴𝑥 → suc 𝐴𝑥))
3736imp 407 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → suc 𝐴𝑥)
38 oveq2 7416 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴))
3938ssiun2s 5051 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 +o 𝑦))
42 vex 3478 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
43 oalim 8531 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4442, 43mpanr1 701 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4544ancoms 459 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥𝐶 ∈ On) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4645adantlr 713 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4746adantlr 713 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = 𝑦𝑥 (𝐶 +o 𝑦))
4841, 47sseqtrrd 4023 . . . . . . . . . . . . . 14 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥))
4948ex 413 . . . . . . . . . . . . 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 7849 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝐵) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
5251exp31 420 . . . . . . . . . 10 (𝐵 ∈ On → (suc 𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
537, 52biimtrid 241 . . . . . . . . 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 418 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (suc 𝐴𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
56 oasuc 8523 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
5756sseq1d 4013 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58 ovex 7441 . . . . . . . . . 10 (𝐶 +o 𝐴) ∈ V
59 sucssel 6459 . . . . . . . . . 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, 60syl6bi 252 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
6261adantlr 713 . . . . . . 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 407 . . . . 5 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
6564an32s 650 . . . 4 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
662, 65mpdan 685 . . 3 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
6766ex 413 . 2 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
6867ancoms 459 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  wss 3948   ciun 4997  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7408   +o coa 8462
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-oadd 8469
This theorem is referenced by:  oaord  8546  oaass  8560  odi  8578  onexomgt  41980  onexoegt  41983  oaltublim  42030  oaordi3  42031  oacl2g  42070  tfsconcatfv2  42080  tfsconcatrn  42082  tfsconcatrev  42088  ofoafg  42094  oaun3lem1  42114  oaun3lem2  42115  oadif1  42120  naddwordnexlem0  42137  naddwordnexlem3  42140  naddwordnexlem4  42142  oaltom  42146
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