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Theorem naddelim 8724
Description: Ordinal less-than is preserved by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
Assertion
Ref Expression
naddelim ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))

Proof of Theorem naddelim
Dummy variables 𝑏 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . . . . . . 9 (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
21eleq1d 2826 . . . . . . . 8 (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
32rspcv 3618 . . . . . . 7 (𝐴𝐵 → (∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
43ad2antlr 727 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) ∧ 𝑥 ∈ On) → (∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
54adantld 490 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) ∧ 𝑥 ∈ On) → ((∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥) → (𝐴 +no 𝐶) ∈ 𝑥))
65ralrimiva 3146 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → ∀𝑥 ∈ On ((∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥) → (𝐴 +no 𝐶) ∈ 𝑥))
7 ovex 7464 . . . . 5 (𝐴 +no 𝐶) ∈ V
87elintrab 4960 . . . 4 ((𝐴 +no 𝐶) ∈ {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)} ↔ ∀𝑥 ∈ On ((∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥) → (𝐴 +no 𝐶) ∈ 𝑥))
96, 8sylibr 234 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐴 +no 𝐶) ∈ {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
10 naddov2 8717 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
11103adant1 1131 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
1211adantr 480 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐵 +no 𝐶) = {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
139, 12eleqtrrd 2844 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶))
1413ex 412 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436   cint 4946  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:  naddel1  8725  naddsuc2  8739
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