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Theorem naddelim 8723
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 2824 . . . . . . . 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 3144 . . . 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 4965 . . . 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 8716 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
11103adant1 1129 . . . 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 2842 . 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 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433   cint 4951  Oncon0 6386  (class class class)co 7431   +no cnadd 8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-nadd 8703
This theorem is referenced by:  naddel1  8724  naddsuc2  8738
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