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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 4114 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | |
2 | eleq1a 2834 | . . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On)) | |
3 | 2 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On)) |
4 | 1, 3 | syl5bi 241 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ On)) |
5 | ssequn2 4117 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴) | |
6 | eleq1a 2834 | . . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On)) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On)) |
8 | 5, 7 | syl5bi 241 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ On)) |
9 | eloni 6276 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
10 | eloni 6276 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
11 | ordtri2or2 6362 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | |
12 | 9, 10, 11 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
13 | 4, 8, 12 | mpjaod 857 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ∪ cun 3885 ⊆ wss 3887 Ord word 6265 Oncon0 6266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-tr 5192 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-ord 6269 df-on 6270 |
This theorem is referenced by: onun2i 6382 nosupinfsep 33935 |
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