Theorem ordtr3OLD 5729

Description: Obsolete proof of ordtr3 5728 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 477	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → Ord 𝐶)
2		ordelord 5704	. . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴)
3	2	adantlr 750	. . . . 5 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → Ord 𝐴)
4		ordtri1 5715	. . . . 5 ⊢ ((Ord 𝐶 ∧ Ord 𝐴) → (𝐶 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐶))
5	1, 3, 4	syl2an2r 875	. . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐶 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐶))
6		ordtr2 5727	. . . . . . 7 ⊢ ((Ord 𝐶 ∧ Ord 𝐵) → ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ 𝐵))
7	6	ancoms 469	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ 𝐵))
8	7	expcomd 454	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ⊆ 𝐴 → 𝐶 ∈ 𝐵)))
9	8	imp 445	. . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐶 ⊆ 𝐴 → 𝐶 ∈ 𝐵))
10	5, 9	sylbird 250	. . 3 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ 𝐶 → 𝐶 ∈ 𝐵))
11	10	orrd 393	. 2 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))
12	11	ex 450	1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∈ wcel 1987   ⊆ wss 3555  Ord word 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685

This theorem is referenced by: (None)