Theorem ordnbtwn 6341

Description: There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (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 6269	. . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		ordn2lp 6271	. . . 4 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		pm2.24 124	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴))
4		eleq2 2827	. . . . . . 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 954	. . . 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 197	. 2 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
10		elsuci 6317	. . 3 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
11	10	anim2i 616	. 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 843   = wceq 1539   ∈ wcel 2108  Ord word 6250  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-fr 5535  df-we 5537  df-ord 6254  df-suc 6257

This theorem is referenced by:  onnbtwn  6342  ordsucss  7640