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Theorem omun 7872
Description: The union of two finite ordinals is a finite ordinal. (Contributed by Scott Fenton, 15-Mar-2025.)
Assertion
Ref Expression
omun ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵) ∈ ω)

Proof of Theorem omun
StepHypRef Expression
1 ssequn1 4173 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 eleq1a 2820 . . . 4 (𝐵 ∈ ω → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ ω))
32adantl 481 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ ω))
41, 3biimtrid 241 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴𝐵) ∈ ω))
5 ssequn2 4176 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
6 eleq1a 2820 . . . 4 (𝐴 ∈ ω → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ ω))
76adantr 480 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ ω))
85, 7biimtrid 241 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝐴 → (𝐴𝐵) ∈ ω))
9 nnord 7857 . . 3 (𝐴 ∈ ω → Ord 𝐴)
10 nnord 7857 . . 3 (𝐵 ∈ ω → Ord 𝐵)
11 ordtri2or2 6454 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
129, 10, 11syl2an 595 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))
134, 8, 12mpjaod 857 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 844   = wceq 1533  wcel 2098  cun 3939  wss 3941  Ord word 6354  ωcom 7849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-om 7850
This theorem is referenced by:  precsexlem10  28033
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