Theorem ordtri2or3 6468

Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6467. (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 6467	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
2		dfss 3965	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
3		sseqin2 4213	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
4		eqcom 2733	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 ↔ 𝐵 = (𝐴 ∩ 𝐵))
5	3, 4	bitri 274	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐴 ∩ 𝐵))
6	2, 5	orbi12i 912	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
7	1, 6	sylib 217	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 845   = wceq 1534   ∩ cin 3945   ⊆ wss 3946  Ord word 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-tr 5263  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6371

This theorem is referenced by:  ordelinel  6469