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Mirrors > Home > MPE Home > Th. List > onun2 | Structured version Visualization version GIF version |
Description: The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.) |
Ref | Expression |
---|---|
onun2 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 4070 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | |
2 | eleq1a 2828 | . . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On)) | |
3 | 2 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On)) |
4 | 1, 3 | syl5bi 245 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ On)) |
5 | ssequn2 4073 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴) | |
6 | eleq1a 2828 | . . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On)) | |
7 | 6 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On)) |
8 | 5, 7 | syl5bi 245 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ On)) |
9 | eloni 6182 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
10 | eloni 6182 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
11 | ordtri2or2 6268 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | |
12 | 9, 10, 11 | syl2an 599 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
13 | 4, 8, 12 | mpjaod 859 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ∪ cun 3841 ⊆ wss 3843 Ord word 6171 Oncon0 6172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-ord 6175 df-on 6176 |
This theorem is referenced by: onun2i 6288 nosupinfsep 33578 |
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