Theorem onsucunipr 43385

Description: The successor to the union of any pair of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)

Assertion

Ref	Expression
onsucunipr	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})

Proof of Theorem onsucunipr

Step	Hyp	Ref	Expression
1		ssequn1 4186	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		suceq 6450	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → suc (𝐴 ∪ 𝐵) = suc 𝐵)
3	1, 2	sylbi 217	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → suc (𝐴 ∪ 𝐵) = suc 𝐵)
4	3	adantl 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc (𝐴 ∪ 𝐵) = suc 𝐵)
5		onsucwordi 43301	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))
6	5	imp 406	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc 𝐴 ⊆ suc 𝐵)
7		ssequn1 4186	. . . . 5 ⊢ (suc 𝐴 ⊆ suc 𝐵 ↔ (suc 𝐴 ∪ suc 𝐵) = suc 𝐵)
8	6, 7	sylib 218	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (suc 𝐴 ∪ suc 𝐵) = suc 𝐵)
9	4, 8	eqtr4d 2780	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
10		ssequn2 4189	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11		suceq 6450	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → suc (𝐴 ∪ 𝐵) = suc 𝐴)
12	10, 11	sylbi 217	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → suc (𝐴 ∪ 𝐵) = suc 𝐴)
13	12	adantl 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc (𝐴 ∪ 𝐵) = suc 𝐴)
14		onsucwordi 43301	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 → suc 𝐵 ⊆ suc 𝐴))
15	14	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → suc 𝐵 ⊆ suc 𝐴))
16	15	imp 406	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc 𝐵 ⊆ suc 𝐴)
17		ssequn2 4189	. . . . 5 ⊢ (suc 𝐵 ⊆ suc 𝐴 ↔ (suc 𝐴 ∪ suc 𝐵) = suc 𝐴)
18	16, 17	sylib 218	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (suc 𝐴 ∪ suc 𝐵) = suc 𝐴)
19	13, 18	eqtr4d 2780	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
20		eloni 6394	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		eloni 6394	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
22		ordtri2or2 6483	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23	20, 21, 22	syl2an 596	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
24	9, 19, 23	mpjaodan 961	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
25		uniprg 4923	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
26		suceq 6450	. . 3 ⊢ (∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
27	25, 26	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
28		onsuc 7831	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
29		onsuc 7831	. . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
30		uniprg 4923	. . 3 ⊢ ((suc 𝐴 ∈ On ∧ suc 𝐵 ∈ On) → ∪ {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
31	28, 29, 30	syl2an 596	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
32	24, 27, 31	3eqtr4d 2787	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ∪ cun 3949   ⊆ wss 3951  {cpr 4628  ∪ cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390

This theorem is referenced by:  onsucunitp  43386