Theorem naddelim 8703

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) → (𝐴 ∈ 𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))

Proof of Theorem naddelim

Dummy variables 𝑏 𝑐 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7417	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
2	1	eleq1d 2820	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
3	2	rspcv 3602	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
4	3	ad2antlr 727	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ On) → (∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
5	4	adantld 490	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ On) → ((∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥) → (𝐴 +no 𝐶) ∈ 𝑥))
6	5	ralrimiva 3133	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) → ∀𝑥 ∈ On ((∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥) → (𝐴 +no 𝐶) ∈ 𝑥))
7		ovex 7443	. . . . 5 ⊢ (𝐴 +no 𝐶) ∈ V
8	7	elintrab 4941	. . . 4 ⊢ ((𝐴 +no 𝐶) ∈ ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥)} ↔ ∀𝑥 ∈ On ((∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥) → (𝐴 +no 𝐶) ∈ 𝑥))
9	6, 8	sylibr 234	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐴 +no 𝐶) ∈ ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
10		naddov2 8696	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
11	10	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
12	11	adantr 480	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐵 +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝐶 (𝐵 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥)})
13	9, 12	eleqtrrd 2838	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶))
14	13	ex 412	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420  ∩ cint 4927  Oncon0 6357  (class class class)co 7410   +no cnadd 8682

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-frecs 8285  df-nadd 8683

This theorem is referenced by:  naddel1  8704  naddsuc2  8718