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Theorem naddelim 8632
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 7365 . . . . . . . . 9 (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
21eleq1d 2823 . . . . . . . 8 (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
32rspcv 3578 . . . . . . 7 (𝐴𝐵 → (∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
43ad2antlr 726 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) ∧ 𝑥 ∈ On) → (∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
54adantld 492 . . . . 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 7391 . . . . 5 (𝐴 +no 𝐶) ∈ V
87elintrab 4922 . . . 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 8626 . . . . 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 482 . . 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 2841 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶))
1413ex 414 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  {crab 3408   cint 4908  Oncon0 6318  (class class class)co 7358   +no cnadd 8612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-frecs 8213  df-nadd 8613
This theorem is referenced by:  naddel1  8633
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