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Mirrors > Home > MPE Home > Th. List > ordunpr | Structured version Visualization version GIF version |
Description: The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
ordunpr | ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6373 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | eloni 6373 | . . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
3 | ordtri2or2 6462 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) | |
4 | 1, 2, 3 | syl2an 595 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) |
5 | 4 | orcomd 870 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶)) |
6 | ssequn2 4179 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵) | |
7 | ssequn1 4176 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶) | |
8 | 6, 7 | orbi12i 913 | . . 3 ⊢ ((𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶) ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶)) |
9 | 5, 8 | sylib 217 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶)) |
10 | unexg 7743 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ V) | |
11 | elprg 4645 | . . 3 ⊢ ((𝐵 ∪ 𝐶) ∈ V → ((𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶} ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶))) | |
12 | 10, 11 | syl 17 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶} ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶))) |
13 | 9, 12 | mpbird 257 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cun 3942 ⊆ wss 3944 {cpr 4626 Ord word 6362 Oncon0 6363 |
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 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
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 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 |
This theorem is referenced by: ordunel 7822 r0weon 10021 |
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