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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 5733 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | eloni 5733 | . . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
3 | ordtri2or2 5823 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) | |
4 | 1, 2, 3 | syl2an 494 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) |
5 | 4 | orcomd 403 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶)) |
6 | ssequn2 3786 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵) | |
7 | ssequn1 3783 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶) | |
8 | 6, 7 | orbi12i 543 | . . 3 ⊢ ((𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶) ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶)) |
9 | 5, 8 | sylib 208 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶)) |
10 | unexg 6959 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ V) | |
11 | elprg 4196 | . . 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 247 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 {cpr 4179 Ord word 5722 Oncon0 5723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 |
This theorem is referenced by: ordunel 7027 r0weon 8835 |
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