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Mirrors > Home > MPE Home > Th. List > ordun | Structured version Visualization version GIF version |
Description: The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.) |
Ref | Expression |
---|---|
ordun | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2800 | . . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵) | |
2 | ordequn 6042 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐴 ∪ 𝐵) = 𝐴 ∨ (𝐴 ∪ 𝐵) = 𝐵))) | |
3 | 1, 2 | mpi 20 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∪ 𝐵) = 𝐴 ∨ (𝐴 ∪ 𝐵) = 𝐵)) |
4 | ordeq 5949 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → (Ord (𝐴 ∪ 𝐵) ↔ Ord 𝐴)) | |
5 | 4 | biimprcd 242 | . . 3 ⊢ (Ord 𝐴 → ((𝐴 ∪ 𝐵) = 𝐴 → Ord (𝐴 ∪ 𝐵))) |
6 | ordeq 5949 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → (Ord (𝐴 ∪ 𝐵) ↔ Ord 𝐵)) | |
7 | 6 | biimprcd 242 | . . 3 ⊢ (Ord 𝐵 → ((𝐴 ∪ 𝐵) = 𝐵 → Ord (𝐴 ∪ 𝐵))) |
8 | 5, 7 | jaao 978 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (((𝐴 ∪ 𝐵) = 𝐴 ∨ (𝐴 ∪ 𝐵) = 𝐵) → Ord (𝐴 ∪ 𝐵))) |
9 | 3, 8 | mpd 15 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 = wceq 1653 ∪ cun 3768 Ord word 5941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-tr 4947 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-ord 5945 |
This theorem is referenced by: ordsucun 7260 r0weon 9122 |
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