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Theorem oaordi 7513
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 5665 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21adantll 746 . . . 4 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → 𝐴 ∈ On)
3 eloni 5650 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
4 ordsucss 6910 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
53, 4syl 17 . . . . . . . 8 (𝐵 ∈ On → (𝐴𝐵 → suc 𝐴𝐵))
65ad2antlr 759 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴𝐵 → suc 𝐴𝐵))
7 sucelon 6909 . . . . . . . . . 10 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8 oveq2 6557 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝐴))
98sseq2d 3596 . . . . . . . . . . . . 13 (𝑥 = suc 𝐴 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)))
109imbi2d 329 . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝑦))
1211sseq2d 3596 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦)))
1312imbi2d 329 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦))))
14 oveq2 6557 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝑦))
1514sseq2d 3596 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
1615imbi2d 329 . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝐵))
1817sseq2d 3596 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
1918imbi2d 329 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
20 ssid 3587 . . . . . . . . . . . . 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 5719 . . . . . . . . . . . . . . . . 17 (𝐶 +𝑜 𝑦) ⊆ suc (𝐶 +𝑜 𝑦)
23 sstr2 3575 . . . . . . . . . . . . . . . . 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 7491 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2625ancoms 468 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2726sseq2d 3596 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦)))
2824, 27syl5ibr 235 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
2928ex 449 . . . . . . . . . . . . . 14 (𝑦 ∈ On → (𝐶 ∈ On → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
3029ad2antrr 758 . . . . . . . . . . . . 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 5736 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
337, 32sylbir 224 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
34 limsuc 6941 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
3534biimpd 218 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 → suc 𝐴𝑥))
3633, 35sylan9r 688 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ suc 𝐴 ∈ On) → (suc 𝐴𝑥 → suc 𝐴𝑥))
3736imp 444 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → suc 𝐴𝑥)
38 oveq2 6557 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝐴 → (𝐶 +𝑜 𝑦) = (𝐶 +𝑜 suc 𝐴))
3938ssiun2s 4500 . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
43 oalim 7499 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4442, 43mpanr1 715 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4544ancoms 468 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥𝐶 ∈ On) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4645adantlr 747 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4746adantlr 747 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4841, 47sseqtr4d 3605 . . . . . . . . . . . . . 14 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥))
4948ex 449 . . . . . . . . . . . . 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 6952 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝐵) → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
5251exp31 628 . . . . . . . . . 10 (𝐵 ∈ On → (suc 𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
537, 52syl5bi 231 . . . . . . . . 9 (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
5453com4r 92 . . . . . . . 8 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
5554imp31 447 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
56 oasuc 7491 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +𝑜 suc 𝐴) = suc (𝐶 +𝑜 𝐴))
5756sseq1d 3595 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) ↔ suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
58 ovex 6577 . . . . . . . . . 10 (𝐶 +𝑜 𝐴) ∈ V
59 sucssel 5736 . . . . . . . . . 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 242 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
6261adantlr 747 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
636, 55, 623syld 58 . . . . . 6 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
6463imp 444 . . . . 5 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
6564an32s 842 . . . 4 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) ∧ 𝐴 ∈ On) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
662, 65mpdan 699 . . 3 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
6766ex 449 . 2 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
6867ancoms 468 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  wss 3540   ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +𝑜 coa 7444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451
This theorem is referenced by:  oaord  7514  oaass  7528  odi  7546
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