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Theorem naddelim 8713
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 7433 . . . . . . . . 9 (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
21eleq1d 2814 . . . . . . . 8 (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
32rspcv 3607 . . . . . . 7 (𝐴𝐵 → (∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
43ad2antlr 725 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) ∧ 𝑥 ∈ On) → (∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
54adantld 489 . . . . 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 3143 . . . 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 7459 . . . . 5 (𝐴 +no 𝐶) ∈ V
87elintrab 4967 . . . 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 233 . . 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 8706 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
11103adant1 1127 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = {𝑥 ∈ On ∣ (∀𝑐𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
1211adantr 479 . . 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 2832 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶))
1413ex 411 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 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  {crab 3430   cint 4953  Oncon0 6374  (class class class)co 7426   +no cnadd 8692
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-frecs 8293  df-nadd 8693
This theorem is referenced by:  naddel1  8714  naddsuc2  42853
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