Theorem ordtr3OLD 5687

Description: Obsolete proof of ordtr3 5686 as of 24-Sep-2021. (Contributed by Mario Carneiro, 30-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ordtr3OLD	⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))

Proof of Theorem ordtr3OLD

Step	Hyp	Ref	Expression
1		simpr 476	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → Ord 𝐶)
2		ordelord 5662	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴)
3	2	adantlr 747	. . . . 5 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → Ord 𝐴)
4		ordtri1 5673	. . . . 5 ⊢ ((Ord 𝐶 ∧ Ord 𝐴) → (𝐶 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐶))
5	1, 3, 4	syl2an2r 872	. . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐶 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐶))
6		ordtr2 5685	. . . . . . 7 ⊢ ((Ord 𝐶 ∧ Ord 𝐵) → ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ 𝐵))
7	6	ancoms 468	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ 𝐵))
8	7	expcomd 453	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ⊆ 𝐴 → 𝐶 ∈ 𝐵)))
9	8	imp 444	. . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐶 ⊆ 𝐴 → 𝐶 ∈ 𝐵))
10	5, 9	sylbird 249	. . 3 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ 𝐶 → 𝐶 ∈ 𝐵))
11	10	orrd 392	. 2 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))
12	11	ex 449	1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540  Ord word 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643

This theorem is referenced by: (None)