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Theorem naddelim 8684
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 7411 . . . . . . . . 9 (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
21eleq1d 2812 . . . . . . . 8 (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
32rspcv 3602 . . . . . . 7 (𝐴𝐵 → (∀𝑏𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
43ad2antlr 724 . . . . . 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 3140 . . . 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 7437 . . . . 5 (𝐴 +no 𝐶) ∈ V
87elintrab 4957 . . . 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 8677 . . . . 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 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 2830 . 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 1084   = wceq 1533  wcel 2098  wral 3055  {crab 3426   cint 4943  Oncon0 6357  (class class class)co 7404   +no cnadd 8663
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 7721
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 6293  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-frecs 8264  df-nadd 8664
This theorem is referenced by:  naddel1  8685  naddsuc2  42701
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