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Theorem nadd2rabtr 42600
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 1211 . . . . . . 7 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → Ord 𝐴)
2 simplr 766 . . . . . . 7 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦𝐴)
3 ordelss 6380 . . . . . . 7 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
41, 2, 3syl2anc 583 . . . . . 6 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦𝐴)
5 simpll3 1213 . . . . . . . 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 773 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑦𝐴)
10 ordelon 6388 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
118, 9, 10syl2anc 583 . . . . . . . . . . 11 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑦 ∈ On)
12 onelon 6389 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
1311, 7, 12syl2anc 583 . . . . . . . . . 10 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑥 ∈ On)
14 simpll2 1212 . . . . . . . . . . 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 8693 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝑦 ↔ (𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦)))
1713, 11, 15, 16syl3anc 1370 . . . . . . . . 9 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → (𝑥𝑦 ↔ (𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦)))
187, 17mpbid 231 . . . . . . . 8 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → (𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦))
19 simplr 766 . . . . . . . 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 6410 . . . . . . 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 4071 . . . . 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 3145 . . 3 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → ∀𝑦𝐴 ((𝐵 +no 𝑦) ∈ 𝐶𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}))
26 oveq2 7420 . . . . 5 (𝑥 = 𝑦 → (𝐵 +no 𝑥) = (𝐵 +no 𝑦))
2726eleq1d 2817 . . . 4 (𝑥 = 𝑦 → ((𝐵 +no 𝑥) ∈ 𝐶 ↔ (𝐵 +no 𝑦) ∈ 𝐶))
2827ralrab 3689 . . 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 233 . 2 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → ∀𝑦 ∈ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
30 dftr3 5271 . 2 (Tr {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ↔ ∀𝑦 ∈ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
3129, 30sylibr 233 1 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086  wcel 2105  wral 3060  {crab 3431  wss 3948  Tr wtr 5265  Ord word 6363  Oncon0 6364  (class class class)co 7412   +no cnadd 8670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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-int 4951  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-se 5632  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-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 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-frecs 8272  df-nadd 8671
This theorem is referenced by:  nadd2rabord  42601  nadd1rabtr  42604
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