Theorem onsucunitp 44333

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

Assertion

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

Proof of Theorem onsucunitp

Step	Hyp	Ref	Expression
1		onun2 6466	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
2		onsucunipr 44332	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
3	1, 2	sylan 592	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
4		uniprg 4883	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5	4	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6		unisng 4885	. . . . . . 7 ⊢ (𝐶 ∈ On → ∪ {𝐶} = 𝐶)
7	6	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐶} = 𝐶)
8	5, 7	uneq12d 4116	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (∪ {𝐴, 𝐵} ∪ ∪ {𝐶}) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
9		df-tp 4589	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
10	9	unieqi 4879	. . . . . . 7 ⊢ ∪ {𝐴, 𝐵, 𝐶} = ∪ ({𝐴, 𝐵} ∪ {𝐶})
11		uniun 4890	. . . . . . 7 ⊢ ∪ ({𝐴, 𝐵} ∪ {𝐶}) = (∪ {𝐴, 𝐵} ∪ ∪ {𝐶})
12	10, 11	eqtri 2784	. . . . . 6 ⊢ ∪ {𝐴, 𝐵, 𝐶} = (∪ {𝐴, 𝐵} ∪ ∪ {𝐶})
13	12	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐴, 𝐵, 𝐶} = (∪ {𝐴, 𝐵} ∪ ∪ {𝐶}))
14		uniprg 4883	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
15	1, 14	sylan 592	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
16	8, 13, 15	3eqtr4d 2806	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐴, 𝐵, 𝐶} = ∪ {(𝐴 ∪ 𝐵), 𝐶})
17		suceq 6424	. . . 4 ⊢ (∪ {𝐴, 𝐵, 𝐶} = ∪ {(𝐴 ∪ 𝐵), 𝐶} → suc ∪ {𝐴, 𝐵, 𝐶} = suc ∪ {(𝐴 ∪ 𝐵), 𝐶})
18	16, 17	syl 18	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = suc ∪ {(𝐴 ∪ 𝐵), 𝐶})
19		df-tp 4589	. . . . . 6 ⊢ {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
20	19	unieqi 4879	. . . . 5 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
21		uniun 4890	. . . . 5 ⊢ ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶}) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
22	20, 21	eqtri 2784	. . . 4 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
23		onsuc 7813	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ On → suc (𝐴 ∪ 𝐵) ∈ On)
24	1, 23	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) ∈ On)
25		onsuc 7813	. . . . . 6 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
26		uniprg 4883	. . . . . 6 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ On ∧ suc 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶))
27	24, 25, 26	syl2an 608	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶))
28		suceq 6424	. . . . . . . . 9 ⊢ (∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
29	4, 28	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
30		onsucunipr 44332	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})
31	29, 30	eqtr3d 2798	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) = ∪ {suc 𝐴, suc 𝐵})
32	31	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc (𝐴 ∪ 𝐵) = ∪ {suc 𝐴, suc 𝐵})
33		unisng 4885	. . . . . . . . 9 ⊢ (suc 𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
34	25, 33	syl 18	. . . . . . . 8 ⊢ (𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
35	34	eqcomd 2767	. . . . . . 7 ⊢ (𝐶 ∈ On → suc 𝐶 = ∪ {suc 𝐶})
36	35	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc 𝐶 = ∪ {suc 𝐶})
37	32, 36	uneq12d 4116	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
38	27, 37	eqtrd 2796	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
39	22, 38	eqtr4id 2815	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
40	3, 18, 39	3eqtr4d 2806	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})
41	40	3impa 1127	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∪ cun 3897  {csn 4584  {cpr 4586  {ctp 4588  ∪ cuni 4867  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-tp 4589  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: (None)