Theorem onsucunitp 43819

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 43818	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
3	1, 2	sylan 581	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {(𝐴 ∪ 𝐵), 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
4		uniprg 4867	. . . . . . 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 4869	. . . . . . 7 ⊢ (𝐶 ∈ On → ∪ {𝐶} = 𝐶)
7	6	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {𝐶} = 𝐶)
8	5, 7	uneq12d 4110	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (∪ {𝐴, 𝐵} ∪ ∪ {𝐶}) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
9		df-tp 4573	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
10	9	unieqi 4863	. . . . . . 7 ⊢ ∪ {𝐴, 𝐵, 𝐶} = ∪ ({𝐴, 𝐵} ∪ {𝐶})
11		uniun 4874	. . . . . . 7 ⊢ ∪ ({𝐴, 𝐵} ∪ {𝐶}) = (∪ {𝐴, 𝐵} ∪ ∪ {𝐶})
12	10, 11	eqtri 2760	. . . . . 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 4867	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
15	1, 14	sylan 581	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {(𝐴 ∪ 𝐵), 𝐶} = ((𝐴 ∪ 𝐵) ∪ 𝐶))
16	8, 13, 15	3eqtr4d 2782	. . . 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 4573	. . . . . 6 ⊢ {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
20	19	unieqi 4863	. . . . 5 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
21		uniun 4874	. . . . 5 ⊢ ∪ ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶}) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
22	20, 21	eqtri 2760	. . . 4 ⊢ ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶})
23		onsuc 7757	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ On → suc (𝐴 ∪ 𝐵) ∈ On)
24	1, 23	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) ∈ On)
25		onsuc 7757	. . . . . 6 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
26		uniprg 4867	. . . . . 6 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ On ∧ suc 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶))
27	24, 25, 26	syl2an 597	. . . . 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 43818	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})
31	29, 30	eqtr3d 2774	. . . . . . 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 4869	. . . . . . . . 9 ⊢ (suc 𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
34	25, 33	syl 17	. . . . . . . 8 ⊢ (𝐶 ∈ On → ∪ {suc 𝐶} = suc 𝐶)
35	34	eqcomd 2743	. . . . . . 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 4110	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (suc (𝐴 ∪ 𝐵) ∪ suc 𝐶) = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
38	27, 37	eqtrd 2772	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶} = (∪ {suc 𝐴, suc 𝐵} ∪ ∪ {suc 𝐶}))
39	22, 38	eqtr4id 2791	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ∪ {suc 𝐴, suc 𝐵, suc 𝐶} = ∪ {suc (𝐴 ∪ 𝐵), suc 𝐶})
40	3, 18, 39	3eqtr4d 2782	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})
41	40	3impa 1110	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∪ cun 3888  {csn 4568  {cpr 4570  {ctp 4572  ∪ cuni 4851  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by: (None)