Theorem ordnbtwn 6274

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 6202	. . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		ordn2lp 6204	. . . 4 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		pm2.24 124	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴))
4		eleq2 2899	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐴))
5	4	biimpac 481	. . . . . 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 200	. 2 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
10		elsuci 6250	. . 3 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
11	10	anim2i 618	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
12	9, 11	nsyl 142	1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1531   ∈ wcel 2108  Ord word 6183  suc csuc 6186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-fr 5507  df-we 5509  df-ord 6187  df-suc 6190

This theorem is referenced by:  onnbtwn  6275  ordsucss  7525