Theorem ordnbtwnOLD 5734

Description: Obsolete proof of ordnbtwn 5733 as of 24-Sep-2021. (Contributed by NM, 21-Jun-1998.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ordnbtwnOLD	⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))

Proof of Theorem ordnbtwnOLD

Step	Hyp	Ref	Expression
1		ordn2lp 5660	. . 3 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
2		ordirr 5658	. . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
3		ioran 510	. . 3 ⊢ (¬ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴) ↔ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∧ ¬ 𝐴 ∈ 𝐴))
4	1, 2, 3	sylanbrc 695	. 2 ⊢ (Ord 𝐴 → ¬ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴))
5		elsuci 5708	. . . . 5 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
6	5	anim2i 591	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
7		andi 907	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ (𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴)))
8	6, 7	sylib 207	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ (𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴)))
9		eleq2 2677	. . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐴))
10	9	biimpac 502	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴) → 𝐴 ∈ 𝐴)
11	10	orim2i 539	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ (𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴)) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴))
12	8, 11	syl 17	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴))
13	4, 12	nsyl 134	1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ord word 5639  suc csuc 5642

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-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-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-fr 4997  df-we 4999  df-ord 5643  df-suc 5646

This theorem is referenced by: (None)