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Theorem onun2 4216
 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 3522 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2 prexg 3947 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3 ssonuni 4214 . . . 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 3595 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
65eleq1d 2106 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( {𝐴, 𝐵} ∈ On ↔ (𝐴𝐵) ∈ On))
74, 6sylibd 138 . 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 97   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915   ⊆ wss 2917  {cpr 3376  ∪ cuni 3580  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by:  onun2i  4217  rdgon  5973
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