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Theorem oaordi 8583
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 6411 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21adantll 714 . . . 4 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → 𝐴 ∈ On)
3 eloni 6396 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
4 ordsucss 7838 . . . . . . . . 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 7837 . . . . . . . . . 10 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8 oveq2 7439 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
98sseq2d 4028 . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
1211sseq2d 4028 . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
1514sseq2d 4028 . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
1817sseq2d 4028 . . . . . . . . . . . . 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 4018 . . . . . . . . . . . . 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 6466 . . . . . . . . . . . . . . . . 17 (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
23 sstr2 4002 . . . . . . . . . . . . . . . . 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 8561 . . . . . . . . . . . . . . . . . 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 4028 . . . . . . . . . . . . . . . 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 6481 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
337, 32sylbir 235 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
34 limsuc 7870 . . . . . . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴))
3938ssiun2s 5053 . . . . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
43 oalim 8569 . . . . . . . . . . . . . . . . . . 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 4037 . . . . . . . . . . . . . 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 7882 . . . . . . . . . . 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 8561 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
5756sseq1d 4027 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58 ovex 7464 . . . . . . . . . 10 (𝐶 +o 𝐴) ∈ V
59 sucssel 6481 . . . . . . . . . 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 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963   ciun 4996  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431   +o coa 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509
This theorem is referenced by:  oaord  8584  oaass  8598  odi  8616  onexomgt  43230  onexoegt  43233  oaltublim  43280  oaordi3  43281  oacl2g  43320  tfsconcatfv2  43330  tfsconcatrn  43332  tfsconcatrev  43338  ofoafg  43344  oaun3lem1  43364  oaun3lem2  43365  oadif1  43370  naddwordnexlem0  43386  naddwordnexlem3  43389  naddwordnexlem4  43391  oaltom  43395
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