Theorem ordtri3OLD 5677

Description: Obsolete proof of ordtri3 5676 as of 24-Sep-2021. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ordtri3OLD	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))

Proof of Theorem ordtri3OLD

Step	Hyp	Ref	Expression
1		ordirr 5658	. . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		eleq2 2677	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	2	notbid 307	. . . . . 6 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵))
4	1, 3	syl5ib 233	. . . . 5 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 → ¬ 𝐴 ∈ 𝐵))
5		ordirr 5658	. . . . . 6 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6		eleq2 2677	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵))
7	6	notbid 307	. . . . . 6 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵))
8	5, 7	syl5ibr 235	. . . . 5 ⊢ (𝐴 = 𝐵 → (Ord 𝐵 → ¬ 𝐵 ∈ 𝐴))
9	4, 8	anim12d 584	. . . 4 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
10	9	com12 32	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
11		pm4.56 515	. . 3 ⊢ ((¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴))
12	10, 11	syl6ib 240	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))
13		ordtri3or 5672	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
14		df-3or 1032	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
15	13, 14	sylib 207	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
16		or32 548	. . . 4 ⊢ (((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐴 = 𝐵))
17	15, 16	sylib 207	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐴 = 𝐵))
18	17	ord 391	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) → 𝐴 = 𝐵))
19	12, 18	impbid 201	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  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)