![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ordtri2or3 | Structured version Visualization version GIF version |
Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6037. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ordtri2or3 | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2or2 6037 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | |
2 | dfss 3784 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) | |
3 | sseqin2 4015 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
4 | eqcom 2806 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 ↔ 𝐵 = (𝐴 ∩ 𝐵)) | |
5 | 3, 4 | bitri 267 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐴 ∩ 𝐵)) |
6 | 2, 5 | orbi12i 939 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) |
7 | 1, 6 | sylib 210 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 = wceq 1653 ∩ cin 3768 ⊆ wss 3769 Ord word 5940 |
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 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
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 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-tr 4946 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-ord 5944 |
This theorem is referenced by: ordelinel 6039 |
Copyright terms: Public domain | W3C validator |