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Theorem onsucunitp 43363
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 6494 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
2 onsucunipr 43362 . . . 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 4928 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
54adantr 480 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
6 unisng 4930 . . . . . . 7 (𝐶 ∈ On → {𝐶} = 𝐶)
76adantl 481 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐶} = 𝐶)
85, 7uneq12d 4179 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ( {𝐴, 𝐵} ∪ {𝐶}) = ((𝐴𝐵) ∪ 𝐶))
9 df-tp 4636 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
109unieqi 4924 . . . . . . 7 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
11 uniun 4935 . . . . . . 7 ({𝐴, 𝐵} ∪ {𝐶}) = ( {𝐴, 𝐵} ∪ {𝐶})
1210, 11eqtri 2763 . . . . . 6 {𝐴, 𝐵, 𝐶} = ( {𝐴, 𝐵} ∪ {𝐶})
1312a1i 11 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐴, 𝐵, 𝐶} = ( {𝐴, 𝐵} ∪ {𝐶}))
14 uniprg 4928 . . . . . 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 2785 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {𝐴, 𝐵, 𝐶} = {(𝐴𝐵), 𝐶})
17 suceq 6452 . . . 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 4636 . . . . . 6 {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
2019unieqi 4924 . . . . 5 {suc 𝐴, suc 𝐵, suc 𝐶} = ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
21 uniun 4935 . . . . 5 ({suc 𝐴, suc 𝐵} ∪ {suc 𝐶}) = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
2220, 21eqtri 2763 . . . 4 {suc 𝐴, suc 𝐵, suc 𝐶} = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶})
23 onsuc 7831 . . . . . . 7 ((𝐴𝐵) ∈ On → suc (𝐴𝐵) ∈ On)
241, 23syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴𝐵) ∈ On)
25 onsuc 7831 . . . . . 6 (𝐶 ∈ On → suc 𝐶 ∈ On)
26 uniprg 4928 . . . . . 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 6452 . . . . . . . . 9 ( {𝐴, 𝐵} = (𝐴𝐵) → suc {𝐴, 𝐵} = suc (𝐴𝐵))
294, 28syl 17 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc {𝐴, 𝐵} = suc (𝐴𝐵))
30 onsucunipr 43362 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc {𝐴, 𝐵} = {suc 𝐴, suc 𝐵})
3129, 30eqtr3d 2777 . . . . . . 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 4930 . . . . . . . . 9 (suc 𝐶 ∈ On → {suc 𝐶} = suc 𝐶)
3425, 33syl 17 . . . . . . . 8 (𝐶 ∈ On → {suc 𝐶} = suc 𝐶)
3534eqcomd 2741 . . . . . . 7 (𝐶 ∈ On → suc 𝐶 = {suc 𝐶})
3635adantl 481 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → suc 𝐶 = {suc 𝐶})
3732, 36uneq12d 4179 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (suc (𝐴𝐵) ∪ suc 𝐶) = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶}))
3827, 37eqtrd 2775 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {suc (𝐴𝐵), suc 𝐶} = ( {suc 𝐴, suc 𝐵} ∪ {suc 𝐶}))
3922, 38eqtr4id 2794 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → {suc 𝐴, suc 𝐵, suc 𝐶} = {suc (𝐴𝐵), suc 𝐶})
403, 18, 393eqtr4d 2785 . 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 1537  wcel 2106  cun 3961  {csn 4631  {cpr 4633  {ctp 4635   cuni 4912  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by: (None)
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