Theorem onsucunitp 43825

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 6427	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
2		onsucunipr 43824	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
3	1, 2	sylan 586	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
4		uniprg 4861	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5	4	adantr 481	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6		unisng 4863	. . . . . . 7 ⊢ (𝐶 ∈ On → ∪ {𝐶} = 𝐶)
7	6	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐶} = 𝐶)
8	5, 7	uneq12d 4106	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (∪ {𝐴, 𝐵} ∪ ∪ {𝐶}) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
9		df-tp 4567	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
10	9	unieqi 4857	. . . . . . 7 ⊢ ∪ {𝐴, 𝐵, 𝐶} = ∪ ({𝐴, 𝐵} ∪ {𝐶})
11		uniun 4868	. . . . . . 7 ⊢ ∪ ({𝐴, 𝐵} ∪ {𝐶}) = (∪ {𝐴, 𝐵} ∪ ∪ {𝐶})
12	10, 11	eqtri 2763	. . . . . 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 4861	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
15	1, 14	sylan 586	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
16	8, 13, 15	3eqtr4d 2785	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐴, 𝐵, 𝐶} = ∪ {(𝐴 ∪ 𝐵), 𝐶})
17		suceq 6385	. . . 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 4567	. . . . . 6 ⊢ {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
20	19	unieqi 4857	. . . . 5 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
21		uniun 4868	. . . . 5 ⊢ ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶}) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
22	20, 21	eqtri 2763	. . . 4 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
23		onsuc 7760	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ On → suc (𝐴 ∪ 𝐵) ∈ On)
24	1, 23	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) ∈ On)
25		onsuc 7760	. . . . . 6 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
26		uniprg 4861	. . . . . 6 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ On ∧ suc 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶))
27	24, 25, 26	syl2an 602	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶))
28		suceq 6385	. . . . . . . . 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 43824	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})
31	29, 30	eqtr3d 2777	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) = ∪ {suc 𝐴, suc 𝐵})
32	31	adantr 481	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc (𝐴 ∪ 𝐵) = ∪ {suc 𝐴, suc 𝐵})
33		unisng 4863	. . . . . . . . 9 ⊢ (suc 𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
34	25, 33	syl 17	. . . . . . . 8 ⊢ (𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
35	34	eqcomd 2746	. . . . . . 7 ⊢ (𝐶 ∈ On → suc 𝐶 = ∪ {suc 𝐶})
36	35	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc 𝐶 = ∪ {suc 𝐶})
37	32, 36	uneq12d 4106	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
38	27, 37	eqtrd 2775	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
39	22, 38	eqtr4id 2794	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
40	3, 18, 39	3eqtr4d 2785	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})
41	40	3impa 1115	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∪ cun 3888  {csn 4562  {cpr 4564  {ctp 4566  ∪ cuni 4845  Oncon0 6317  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by: (None)