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| Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6483. (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 6483 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | |
| 2 | dfss 3970 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) | |
| 3 | sseqin2 4223 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
| 4 | eqcom 2744 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 ↔ 𝐵 = (𝐴 ∩ 𝐵)) | |
| 5 | 3, 4 | bitri 275 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐴 ∩ 𝐵)) | 
| 6 | 2, 5 | orbi12i 915 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) | 
| 7 | 1, 6 | sylib 218 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 Ord word 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 | 
| This theorem is referenced by: ordelinel 6485 | 
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