Theorem naddelim 8607

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 7359	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
2	1	eleq1d 2818	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
3	2	rspcv 3569	. . . . . . 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 3125	. . . 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 7385	. . . . 5 ⊢ (𝐴 +no 𝐶) ∈ V
8	7	elintrab 4910	. . . 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 8600	. . . . 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 2836	. 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 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396  ∩ cint 4897  Oncon0 6311  (class class class)co 7352   +no cnadd 8586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-frecs 8217  df-nadd 8587

This theorem is referenced by:  naddel1  8608  naddsuc2  8622