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Theorem onsucunitp 43391
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
StepHypRef Expression
1 onun2 6491 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
2 onsucunipr 43390 . . . 4 (((𝐴𝐵) ∈ On ∧ 𝐶 ∈ On) → suc {(𝐴𝐵), 𝐶} = {suc (𝐴𝐵), suc 𝐶})
31, 2sylan 580 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc {(𝐴𝐵), 𝐶} = {suc (𝐴𝐵), suc 𝐶})
4 uniprg 4922 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
54adantr 480 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
6 unisng 4924 . . . . . . 7 (𝐶 ∈ On → {𝐶} = 𝐶)
76adantl 481 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐶} = 𝐶)
85, 7uneq12d 4168 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ( {𝐴, 𝐵} ∪ {𝐶}) = ((𝐴𝐵) ∪ 𝐶))
9 df-tp 4630 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
109unieqi 4918 . . . . . . 7 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
11 uniun 4929 . . . . . . 7 ({𝐴, 𝐵} ∪ {𝐶}) = ( {𝐴, 𝐵} ∪ {𝐶})
1210, 11eqtri 2764 . . . . . 6 {𝐴, 𝐵, 𝐶} = ( {𝐴, 𝐵} ∪ {𝐶})
1312a1i 11 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐴, 𝐵, 𝐶} = ( {𝐴, 𝐵} ∪ {𝐶}))
14 uniprg 4922 . . . . . 6 (((𝐴𝐵) ∈ On ∧ 𝐶 ∈ On) → {(𝐴𝐵), 𝐶} = ((𝐴𝐵) ∪ 𝐶))
151, 14sylan 580 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {(𝐴𝐵), 𝐶} = ((𝐴𝐵) ∪ 𝐶))
168, 13, 153eqtr4d 2786 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐴, 𝐵, 𝐶} = {(𝐴𝐵), 𝐶})
17 suceq 6449 . . . 4 ( {𝐴, 𝐵, 𝐶} = {(𝐴𝐵), 𝐶} → suc {𝐴, 𝐵, 𝐶} = suc {(𝐴𝐵), 𝐶})
1816, 17syl 17 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc {𝐴, 𝐵, 𝐶} = suc {(𝐴𝐵), 𝐶})
19 df-tp 4630 . . . . . 6 {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
2019unieqi 4918 . . . . 5 {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
21 uniun 4929 . . . . 5 ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶}) = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
2220, 21eqtri 2764 . . . 4 {suc 𝐴, suc 𝐵, suc 𝐶} = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
23 onsuc 7832 . . . . . . 7 ((𝐴𝐵) ∈ On → suc (𝐴𝐵) ∈ On)
241, 23syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴𝐵) ∈ On)
25 onsuc 7832 . . . . . 6 (𝐶 ∈ On → suc 𝐶 ∈ On)
26 uniprg 4922 . . . . . 6 ((suc (𝐴𝐵) ∈ On ∧ suc 𝐶 ∈ On) → {suc (𝐴𝐵), suc 𝐶} = (suc (𝐴𝐵) ∪ suc 𝐶))
2724, 25, 26syl2an 596 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {suc (𝐴𝐵), suc 𝐶} = (suc (𝐴𝐵) ∪ suc 𝐶))
28 suceq 6449 . . . . . . . . 9 ( {𝐴, 𝐵} = (𝐴𝐵) → suc {𝐴, 𝐵} = suc (𝐴𝐵))
294, 28syl 17 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc {𝐴, 𝐵} = suc (𝐴𝐵))
30 onsucunipr 43390 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc {𝐴, 𝐵} = {suc 𝐴, suc 𝐵})
3129, 30eqtr3d 2778 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴𝐵) = {suc 𝐴, suc 𝐵})
3231adantr 480 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc (𝐴𝐵) = {suc 𝐴, suc 𝐵})
33 unisng 4924 . . . . . . . . 9 (suc 𝐶 ∈ On → {suc 𝐶} = suc 𝐶)
3425, 33syl 17 . . . . . . . 8 (𝐶 ∈ On → {suc 𝐶} = suc 𝐶)
3534eqcomd 2742 . . . . . . 7 (𝐶 ∈ On → suc 𝐶 = {suc 𝐶})
3635adantl 481 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc 𝐶 = {suc 𝐶})
3732, 36uneq12d 4168 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (suc (𝐴𝐵) ∪ suc 𝐶) = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶}))
3827, 37eqtrd 2776 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {suc (𝐴𝐵), suc 𝐶} = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶}))
3922, 38eqtr4id 2795 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {suc 𝐴, suc 𝐵, suc 𝐶} = {suc (𝐴𝐵), suc 𝐶})
403, 18, 393eqtr4d 2786 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc {𝐴, 𝐵, 𝐶} = {suc 𝐴, suc 𝐵, suc 𝐶})
41403impa 1109 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc {𝐴, 𝐵, 𝐶} = {suc 𝐴, suc 𝐵, suc 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  cun 3948  {csn 4625  {cpr 4627  {ctp 4629   cuni 4906  Oncon0 6383  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-suc 6389
This theorem is referenced by: (None)
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