Theorem ordtri3OLD 5760

Description: Obsolete proof of ordtri3 5759 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 5741	. . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		eleq2 2690	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	2	notbid 308	. . . . . 6 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵))
4	1, 3	syl5ib 234	. . . . 5 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 → ¬ 𝐴 ∈ 𝐵))
5		ordirr 5741	. . . . . 6 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6		eleq2 2690	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵))
7	6	notbid 308	. . . . . 6 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵))
8	5, 7	syl5ibr 236	. . . . 5 ⊢ (𝐴 = 𝐵 → (Ord 𝐵 → ¬ 𝐵 ∈ 𝐴))
9	4, 8	anim12d 586	. . . 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		pm4.56 516	. . 3 ⊢ ((¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴))
12	10, 11	syl6ib 241	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))
13		ordtri3or 5755	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
14		df-3or 1038	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
15	13, 14	sylib 208	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
16		or32 549	. . . 4 ⊢ (((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐴 = 𝐵))
17	15, 16	sylib 208	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐴 = 𝐵))
18	17	ord 392	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) → 𝐴 = 𝐵))
19	12, 18	impbid 202	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990  Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by: (None)