Theorem onsucunipr 44332

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 4132	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		suceq 6424	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → suc (𝐴 ∪ 𝐵) = suc 𝐵)
3	1, 2	sylbi 220	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → suc (𝐴 ∪ 𝐵) = suc 𝐵)
4	3	adantl 487	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc (𝐴 ∪ 𝐵) = suc 𝐵)
5		onsucwordi 44248	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))
6	5	imp 412	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc 𝐴 ⊆ suc 𝐵)
7		ssequn1 4132	. . . . 5 ⊢ (suc 𝐴 ⊆ suc 𝐵 ↔ (suc 𝐴 ∪ suc 𝐵) = suc 𝐵)
8	6, 7	sylib 221	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (suc 𝐴 ∪ suc 𝐵) = suc 𝐵)
9	4, 8	eqtr4d 2799	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
10		ssequn2 4135	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11		suceq 6424	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → suc (𝐴 ∪ 𝐵) = suc 𝐴)
12	10, 11	sylbi 220	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → suc (𝐴 ∪ 𝐵) = suc 𝐴)
13	12	adantl 487	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc (𝐴 ∪ 𝐵) = suc 𝐴)
14		onsucwordi 44248	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 → suc 𝐵 ⊆ suc 𝐴))
15	14	ancoms 464	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → suc 𝐵 ⊆ suc 𝐴))
16	15	imp 412	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc 𝐵 ⊆ suc 𝐴)
17		ssequn2 4135	. . . . 5 ⊢ (suc 𝐵 ⊆ suc 𝐴 ↔ (suc 𝐴 ∪ suc 𝐵) = suc 𝐴)
18	16, 17	sylib 221	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → (suc 𝐴 ∪ suc 𝐵) = suc 𝐴)
19	13, 18	eqtr4d 2799	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
20		eloni 6365	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		eloni 6365	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
22		ordtri2or2 6457	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23	20, 21, 22	syl2an 608	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
24	9, 19, 23	mpjaodan 973	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
25		uniprg 4883	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
26		suceq 6424	. . 3 ⊢ (∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
27	25, 26	syl 18	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
28		onsuc 7813	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
29		onsuc 7813	. . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
30		uniprg 4883	. . 3 ⊢ ((suc 𝐴 ∈ On ∧ suc 𝐵 ∈ On) → ∪ {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
31	28, 29, 30	syl2an 608	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
32	24, 27, 31	3eqtr4d 2806	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ∪ cun 3897   ⊆ wss 3899  {cpr 4586  ∪ cuni 4867  Ord word 6354  Oncon0 6355  suc csuc 6357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6358  df-on 6359  df-suc 6361

This theorem is used by:  onsucunitp  44333