Theorem ordtri2or3 6434

Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6433. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
ordtri2or3	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))

Proof of Theorem ordtri2or3

Step	Hyp	Ref	Expression
1		ordtri2or2 6433	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
2		dfss 3933	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
3		sseqin2 4186	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
4		eqcom 2736	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 ↔ 𝐵 = (𝐴 ∩ 𝐵))
5	3, 4	bitri 275	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐴 ∩ 𝐵))
6	2, 5	orbi12i 914	. 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)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∩ cin 3913   ⊆ wss 3914  Ord word 6331

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335

This theorem is referenced by:  ordelinel  6435