Theorem onsucunipr 44082

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 4140	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		suceq 6431	. . . . . 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 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc (𝐴 ∪ 𝐵) = suc 𝐵)
5		onsucwordi 43998	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))
6	5	imp 411	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc 𝐴 ⊆ suc 𝐵)
7		ssequn1 4140	. . . . 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 2801	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
10		ssequn2 4143	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
11		suceq 6431	. . . . . 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 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc (𝐴 ∪ 𝐵) = suc 𝐴)
14		onsucwordi 43998	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 → suc 𝐵 ⊆ suc 𝐴))
15	14	ancoms 463	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → suc 𝐵 ⊆ suc 𝐴))
16	15	imp 411	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc 𝐵 ⊆ suc 𝐴)
17		ssequn2 4143	. . . . 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 2801	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
20		eloni 6372	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		eloni 6372	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
22		ordtri2or2 6464	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23	20, 21, 22	syl2an 607	. . 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 4889	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
26		suceq 6431	. . 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 7810	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
29		onsuc 7810	. . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
30		uniprg 4889	. . 3 ⊢ ((suc 𝐴 ∈ On ∧ suc 𝐵 ∈ On) → ∪ {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
31	28, 29, 30	syl2an 607	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
32	24, 27, 31	3eqtr4d 2808	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ∪ cun 3904   ⊆ wss 3906  {cpr 4592  ∪ cuni 4873  Ord word 6361  Oncon0 6362  suc csuc 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366  df-suc 6368

This theorem is referenced by:  onsucunitp  44083