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Theorem onsucunipr 43385
Description: The successor to the union of any pair of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
Assertion
Ref Expression
onsucunipr ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc {𝐴, 𝐵} = {suc 𝐴, suc 𝐵})

Proof of Theorem onsucunipr
StepHypRef Expression
1 ssequn1 4186 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 suceq 6450 . . . . . 6 ((𝐴𝐵) = 𝐵 → suc (𝐴𝐵) = suc 𝐵)
31, 2sylbi 217 . . . . 5 (𝐴𝐵 → suc (𝐴𝐵) = suc 𝐵)
43adantl 481 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → suc (𝐴𝐵) = suc 𝐵)
5 onsucwordi 43301 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))
65imp 406 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → suc 𝐴 ⊆ suc 𝐵)
7 ssequn1 4186 . . . . 5 (suc 𝐴 ⊆ suc 𝐵 ↔ (suc 𝐴 ∪ suc 𝐵) = suc 𝐵)
86, 7sylib 218 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (suc 𝐴 ∪ suc 𝐵) = suc 𝐵)
94, 8eqtr4d 2780 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
10 ssequn2 4189 . . . . . 6 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
11 suceq 6450 . . . . . 6 ((𝐴𝐵) = 𝐴 → suc (𝐴𝐵) = suc 𝐴)
1210, 11sylbi 217 . . . . 5 (𝐵𝐴 → suc (𝐴𝐵) = suc 𝐴)
1312adantl 481 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → suc (𝐴𝐵) = suc 𝐴)
14 onsucwordi 43301 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵𝐴 → suc 𝐵 ⊆ suc 𝐴))
1514ancoms 458 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 → suc 𝐵 ⊆ suc 𝐴))
1615imp 406 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → suc 𝐵 ⊆ suc 𝐴)
17 ssequn2 4189 . . . . 5 (suc 𝐵 ⊆ suc 𝐴 ↔ (suc 𝐴 ∪ suc 𝐵) = suc 𝐴)
1816, 17sylib 218 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → (suc 𝐴 ∪ suc 𝐵) = suc 𝐴)
1913, 18eqtr4d 2780 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
20 eloni 6394 . . . 4 (𝐴 ∈ On → Ord 𝐴)
21 eloni 6394 . . . 4 (𝐵 ∈ On → Ord 𝐵)
22 ordtri2or2 6483 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
2320, 21, 22syl2an 596 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐵𝐴))
249, 19, 23mpjaodan 961 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
25 uniprg 4923 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
26 suceq 6450 . . 3 ( {𝐴, 𝐵} = (𝐴𝐵) → suc {𝐴, 𝐵} = suc (𝐴𝐵))
2725, 26syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc {𝐴, 𝐵} = suc (𝐴𝐵))
28 onsuc 7831 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ On)
29 onsuc 7831 . . 3 (𝐵 ∈ On → suc 𝐵 ∈ On)
30 uniprg 4923 . . 3 ((suc 𝐴 ∈ On ∧ suc 𝐵 ∈ On) → {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
3128, 29, 30syl2an 596 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {suc 𝐴, suc 𝐵} = (suc 𝐴 ∪ suc 𝐵))
3224, 27, 313eqtr4d 2787 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc {𝐴, 𝐵} = {suc 𝐴, suc 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  cun 3949  wss 3951  {cpr 4628   cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  onsucunitp  43386
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