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Theorem nadd2rabtr 42877
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 1209 . . . . . . 7 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → Ord 𝐴)
2 simplr 767 . . . . . . 7 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦𝐴)
3 ordelss 6380 . . . . . . 7 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
41, 2, 3syl2anc 582 . . . . . 6 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦𝐴)
5 simpll3 1211 . . . . . . . 8 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝐶 ∈ On)
65adantr 479 . . . . . . 7 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝐶 ∈ On)
7 simpr 483 . . . . . . . . 9 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑥𝑦)
81adantr 479 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → Ord 𝐴)
9 simpllr 774 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑦𝐴)
10 ordelon 6388 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
118, 9, 10syl2anc 582 . . . . . . . . . . 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 582 . . . . . . . . . 10 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝑥 ∈ On)
14 simpll2 1210 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝐵 ∈ On)
1514adantr 479 . . . . . . . . . 10 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → 𝐵 ∈ On)
16 naddel2 8705 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝑦 ↔ (𝐵 +no 𝑥) ∈ (𝐵 +no 𝑦)))
1713, 11, 15, 16syl3anc 1368 . . . . . . . . 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 767 . . . . . . . 8 (((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) ∧ 𝑥𝑦) → (𝐵 +no 𝑦) ∈ 𝐶)
2018, 19jca 510 . . . . . . 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 4063 . . . . 5 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
2423ex 411 . . . 4 (((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) → ((𝐵 +no 𝑦) ∈ 𝐶𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}))
2524ralrimiva 3136 . . 3 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → ∀𝑦𝐴 ((𝐵 +no 𝑦) ∈ 𝐶𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}))
26 oveq2 7423 . . . . 5 (𝑥 = 𝑦 → (𝐵 +no 𝑥) = (𝐵 +no 𝑦))
2726eleq1d 2810 . . . 4 (𝑥 = 𝑦 → ((𝐵 +no 𝑥) ∈ 𝐶 ↔ (𝐵 +no 𝑦) ∈ 𝐶))
2827ralrab 3681 . . 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 5266 . 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 394  w3a 1084  wcel 2098  wral 3051  {crab 3419  wss 3940  Tr wtr 5260  Ord word 6363  Oncon0 6364  (class class class)co 7415   +no cnadd 8682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-frecs 8283  df-nadd 8683
This theorem is referenced by:  nadd2rabord  42878  nadd1rabtr  42881
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