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Theorem nadd2rabtr 43397
Description: The set of ordinals which have a natural sum less than some ordinal is transitive. (Contributed by RP, 20-Dec-2024.)
Assertion
Ref Expression
nadd2rabtr ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem nadd2rabtr
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpll1 1213 . . . . . . 7 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → Ord 𝐴)
2 simplr 769 . . . . . . 7 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦𝐴)
3 ordelss 6400 . . . . . . 7 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
41, 2, 3syl2anc 584 . . . . . 6 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦𝐴)
5 simpll3 1215 . . . . . . . 8 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝐶 ∈ On)
65adantr 480 . . . . . . 7 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝐶 ∈ On)
7 simpr 484 . . . . . . . . 9 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑥𝑦)
81adantr 480 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → Ord 𝐴)
9 simpllr 776 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑦𝐴)
10 ordelon 6408 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
118, 9, 10syl2anc 584 . . . . . . . . . . 11 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑦 ∈ On)
12 onelon 6409 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
1311, 7, 12syl2anc 584 . . . . . . . . . 10 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑥 ∈ On)
14 simpll2 1214 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝐵 ∈ On)
1514adantr 480 . . . . . . . . . 10 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝐵 ∈ On)
16 naddel2 8726 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝑦 ↔ (𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦)))
1713, 11, 15, 16syl3anc 1373 . . . . . . . . 9 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → (𝑥𝑦 ↔ (𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦)))
187, 17mpbid 232 . . . . . . . 8 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → (𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦))
19 simplr 769 . . . . . . . 8 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → (𝐵 +no 𝑦) ∈ 𝐶)
2018, 19jca 511 . . . . . . 7 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → ((𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦) ∧ (𝐵 +no 𝑦) ∈ 𝐶))
21 ontr1 6430 . . . . . . 7 (𝐶 ∈ On → (((𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → (𝐵 +no 𝑥) ∈ 𝐶))
226, 20, 21sylc 65 . . . . . 6 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → (𝐵 +no 𝑥) ∈ 𝐶)
234, 22ssrabdv 4074 . . . . 5 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
2423ex 412 . . . 4 (((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) → ((𝐵 +no 𝑦) ∈ 𝐶𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}))
2524ralrimiva 3146 . . 3 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → ∀𝑦𝐴 ((𝐵 +no 𝑦) ∈ 𝐶𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}))
26 oveq2 7439 . . . . 5 (𝑥 = 𝑦 → (𝐵 +no 𝑥) = (𝐵 +no 𝑦))
2726eleq1d 2826 . . . 4 (𝑥 = 𝑦 → ((𝐵 +no 𝑥) ∈ 𝐶 ↔ (𝐵 +no 𝑦) ∈ 𝐶))
2827ralrab 3699 . . 3 (∀𝑦 ∈ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ↔ ∀𝑦𝐴 ((𝐵 +no 𝑦) ∈ 𝐶𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}))
2925, 28sylibr 234 . 2 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → ∀𝑦 ∈ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
30 dftr3 5265 . 2 (Tr {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ↔ ∀𝑦 ∈ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
3129, 30sylibr 234 1 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wcel 2108  wral 3061  {crab 3436  wss 3951  Tr wtr 5259  Ord word 6383  Oncon0 6384  (class class class)co 7431   +no cnadd 8703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-nadd 8704
This theorem is referenced by:  nadd2rabord  43398  nadd1rabtr  43401
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