Theorem naddelim 33838

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

Step	Hyp	Ref	Expression
1		oveq1 7282	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
2	1	eleq1d 2823	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
3	2	rspcv 3557	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
4	3	ad2antlr 724	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ On) → (∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
5	4	adantld 491	. . . . 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 3103	. . . 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 7308	. . . . 5 ⊢ (𝐴 +no 𝐶) ∈ V
8	7	elintrab 4891	. . . 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 233	. . 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 33834	. . . . 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 1129	. . . 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 481	. . 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 2842	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶))
14	13	ex 413	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  ∩ cint 4879  Oncon0 6266  (class class class)co 7275   +no cnadd 33824

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-frecs 8097  df-nadd 33825

This theorem is referenced by:  naddel1  33839