Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nadd2rabtr Structured version   Visualization version   GIF version

Theorem nadd2rabtr 42715
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 766 . . . . . . 7 ((((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑦𝐴) ∧ (𝐵 +no 𝑦) ∈ 𝐶) → 𝑦𝐴)
3 ordelss 6374 . . . . . . 7 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
41, 2, 3syl2anc 583 . . . . . 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 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 6382 . . . . . . . . . . . 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 6383 . . . . . . . . . . 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 1210 . . . . . . . . . . 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 8689 . . . . . . . . . 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 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 6404 . . . . . . 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 4066 . . . . 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 3140 . . 3 ((Ord 𝐴𝐵 ∈ On ∧ 𝐶 ∈ On) → ∀𝑦𝐴 ((𝐵 +no 𝑦) ∈ 𝐶𝑦 ⊆ {𝑥𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}))
26 oveq2 7413 . . . . 5 (𝑥 = 𝑦 → (𝐵 +no 𝑥) = (𝐵 +no 𝑦))
2726eleq1d 2812 . . . 4 (𝑥 = 𝑦 → ((𝐵 +no 𝑥) ∈ 𝐶 ↔ (𝐵 +no 𝑦) ∈ 𝐶))
2827ralrab 3684 . . 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 5264 . 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 1084  wcel 2098  wral 3055  {crab 3426  wss 3943  Tr wtr 5258  Ord word 6357  Oncon0 6358  (class class class)co 7405   +no cnadd 8666
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-frecs 8267  df-nadd 8667
This theorem is referenced by:  nadd2rabord  42716  nadd1rabtr  42719
  Copyright terms: Public domain W3C validator