Theorem ordnbtwn 6401

Description: There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. Lemma 1.15 of [Schloeder] p. 2. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.)

Assertion

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

Proof of Theorem ordnbtwn

Step	Hyp	Ref	Expression
1		ordirr 6324	. . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		ordn2lp 6326	. . . 4 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		pm2.24 124	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴))
4		eleq2 2820	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐴))
5	4	biimpac 478	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴) → 𝐴 ∈ 𝐴)
6	5	a1d 25	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴))
7	3, 6	jaodan 959	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴))
8	2, 7	syl5com 31	. . 3 ⊢ (Ord 𝐴 → ((𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → 𝐴 ∈ 𝐴))
9	1, 8	mtod 198	. 2 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
10		elsuci 6375	. . 3 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
11	10	anim2i 617	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
12	9, 11	nsyl 140	1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  Ord word 6305  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-fr 5567  df-we 5569  df-ord 6309  df-suc 6312

This theorem is referenced by:  onnbtwn  6402  ordsucss  7748  ordnexbtwnsuc  43359