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Theorem onun2 4335
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 3617 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2 prexg 4062 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3 ssonuni 4333 . . . 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 3690 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} = (𝐴𝐵))
65eleq1d 2163 . . 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 1445  Vcvv 2633  cun 3011  wss 3013  {cpr 3467   cuni 3675  Oncon0 4214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219
This theorem is referenced by:  onun2i  4336  rdgon  6189
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