Theorem onsucunitp 43405

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 6416	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
2		onsucunipr 43404	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
3	1, 2	sylan 580	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
4		uniprg 4875	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5	4	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6		unisng 4877	. . . . . . 7 ⊢ (𝐶 ∈ On → ∪ {𝐶} = 𝐶)
7	6	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐶} = 𝐶)
8	5, 7	uneq12d 4119	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (∪ {𝐴, 𝐵} ∪ ∪ {𝐶}) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
9		df-tp 4581	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
10	9	unieqi 4871	. . . . . . 7 ⊢ ∪ {𝐴, 𝐵, 𝐶} = ∪ ({𝐴, 𝐵} ∪ {𝐶})
11		uniun 4882	. . . . . . 7 ⊢ ∪ ({𝐴, 𝐵} ∪ {𝐶}) = (∪ {𝐴, 𝐵} ∪ ∪ {𝐶})
12	10, 11	eqtri 2754	. . . . . 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 4875	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
15	1, 14	sylan 580	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
16	8, 13, 15	3eqtr4d 2776	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐴, 𝐵, 𝐶} = ∪ {(𝐴 ∪ 𝐵), 𝐶})
17		suceq 6374	. . . 4 ⊢ (∪ {𝐴, 𝐵, 𝐶} = ∪ {(𝐴 ∪ 𝐵), 𝐶} → suc ∪ {𝐴, 𝐵, 𝐶} = suc ∪ {(𝐴 ∪ 𝐵), 𝐶})
18	16, 17	syl 17	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = suc ∪ {(𝐴 ∪ 𝐵), 𝐶})
19		df-tp 4581	. . . . . 6 ⊢ {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
20	19	unieqi 4871	. . . . 5 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
21		uniun 4882	. . . . 5 ⊢ ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶}) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
22	20, 21	eqtri 2754	. . . 4 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
23		onsuc 7743	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ On → suc (𝐴 ∪ 𝐵) ∈ On)
24	1, 23	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) ∈ On)
25		onsuc 7743	. . . . . 6 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
26		uniprg 4875	. . . . . 6 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ On ∧ suc 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶))
27	24, 25, 26	syl2an 596	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶))
28		suceq 6374	. . . . . . . . 9 ⊢ (∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
29	4, 28	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = suc (𝐴 ∪ 𝐵))
30		onsucunipr 43404	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})
31	29, 30	eqtr3d 2768	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) = ∪ {suc 𝐴, suc 𝐵})
32	31	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc (𝐴 ∪ 𝐵) = ∪ {suc 𝐴, suc 𝐵})
33		unisng 4877	. . . . . . . . 9 ⊢ (suc 𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
34	25, 33	syl 17	. . . . . . . 8 ⊢ (𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
35	34	eqcomd 2737	. . . . . . 7 ⊢ (𝐶 ∈ On → suc 𝐶 = ∪ {suc 𝐶})
36	35	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc 𝐶 = ∪ {suc 𝐶})
37	32, 36	uneq12d 4119	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
38	27, 37	eqtrd 2766	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
39	22, 38	eqtr4id 2785	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
40	3, 18, 39	3eqtr4d 2776	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})
41	40	3impa 1109	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∪ cun 3900  {csn 4576  {cpr 4578  {ctp 4580  ∪ cuni 4859  Oncon0 6306  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310  df-suc 6312

This theorem is referenced by: (None)