onun2 - Metamath Proof Explorer

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Theorem onun2 6494

Description: The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.)

**Assertion**
Ref	Expression
onun2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)

**Proof of Theorem onun2**
Step	Hyp	Ref	Expression
1		ssequn1 4187	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		eleq1a 2859	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On))
3	2	adantl 487	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On))
4	1, 3	biimtrid 245	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ On))
5		ssequn2 4190	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
6		eleq1a 2859	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On))
7	6	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On))
8	5, 7	biimtrid 245	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ On))
9		eloni 6393	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
10		eloni 6393	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
11		ordtri2or2 6485	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
12	9, 10, 11	syl2an 608	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
13	4, 8, 12	mpjaod 874	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∪ cun 4152 Ord word 6382 Oncon0 6383

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5253 ax-pr 5396

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-tr 5217 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-ord 6386 df-on 6387

This theorem is used by: onun2i 6507 nosupinfsep 28122 r0maxss 36016 maxssj0 36019 j0ltalephlem 36020 j0ltaleph 36021 onexlimgt 44448 omabs2 44537 onsucunitp 44578