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Theorem omun 7887
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 4176 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 eleq1a 2824 . . . 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 4179 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
6 eleq1a 2824 . . . 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 7872 . . 3 (𝐴 ∈ ω → Ord 𝐴)
10 nnord 7872 . . 3 (𝐵 ∈ ω → Ord 𝐵)
11 ordtri2or2 6462 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
129, 10, 11syl2an 595 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))
134, 8, 12mpjaod 859 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846   = wceq 1534  wcel 2099  cun 3943  wss 3945  Ord word 6362  ωcom 7864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-om 7865
This theorem is referenced by:  precsexlem10  28107
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