Theorem naddelim 8685

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 7416	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → (𝑏 +no 𝐶) = (𝐴 +no 𝐶))
2	1	eleq1d 2819	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → ((𝑏 +no 𝐶) ∈ 𝑥 ↔ (𝐴 +no 𝐶) ∈ 𝑥))
3	2	rspcv 3609	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
4	3	ad2antlr 726	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ On) → (∀𝑏 ∈ 𝐵 (𝑏 +no 𝐶) ∈ 𝑥 → (𝐴 +no 𝐶) ∈ 𝑥))
5	4	adantld 492	. . . . 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 3147	. . . 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 7442	. . . . 5 ⊢ (𝐴 +no 𝐶) ∈ V
8	7	elintrab 4965	. . . 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 8678	. . . . 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 1131	. . . 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 482	. . 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 2837	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶))
14	13	ex 414	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  ∩ cint 4951  Oncon0 6365  (class class class)co 7409   +no cnadd 8664

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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-frecs 8266  df-nadd 8665

This theorem is referenced by:  naddel1  8686  naddsuc2  42143