Theorem ordtri2or3 6419

Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6418. (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 6418	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
2		dfss 3920	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
3		sseqin2 4175	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
4		eqcom 2743	. . . 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 1541   ∩ cin 3900   ⊆ wss 3901  Ord word 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320

This theorem is referenced by:  ordelinel  6420