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Theorem onun2 4472
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 3736 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2 prexg 4194 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3 ssonuni 4470 . . . 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 3809 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
65eleq1d 2239 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( {𝐴, 𝐵} ∈ On ↔ (𝐴𝐵) ∈ On))
74, 6sylibd 148 . 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 103  wcel 2141  Vcvv 2730  cun 3119  wss 3121  {cpr 3582   cuni 3794  Oncon0 4346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351
This theorem is referenced by:  onun2i  4473  rdgon  6362
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