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Theorem onun2 4269
Description: The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
Assertion
Ref Expression
onun2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)

Proof of Theorem onun2
StepHypRef Expression
1 prssi 3569 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2 prexg 4001 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3 ssonuni 4267 . . . 4 ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} ⊆ On → {𝐴, 𝐵} ∈ On))
42, 3syl 14 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, 𝐵} ⊆ On → {𝐴, 𝐵} ∈ On))
5 uniprg 3642 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
65eleq1d 2151 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( {𝐴, 𝐵} ∈ On ↔ (𝐴𝐵) ∈ On))
74, 6sylibd 147 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, 𝐵} ⊆ On → (𝐴𝐵) ∈ On))
81, 7mpd 13 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  Vcvv 2612  cun 2982  wss 2984  {cpr 3423   cuni 3627  Oncon0 4153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-sn 3428  df-pr 3429  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158
This theorem is referenced by:  onun2i  4270  rdgon  6081
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