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Theorem onun2 4507
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 3765 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2 prexg 4229 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3 ssonuni 4505 . . . 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 3839 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
65eleq1d 2258 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( {𝐴, 𝐵} ∈ On ↔ (𝐴𝐵) ∈ On))
74, 6sylibd 149 . 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 104  wcel 2160  Vcvv 2752  cun 3142  wss 3144  {cpr 3608   cuni 3824  Oncon0 4381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386
This theorem is referenced by:  onun2i  4508  rdgon  6410
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